Student Difficulties in Mathematizing Word Problems in Algebra
ERIC Educational Resources Information Center
Jupri, Al; Drijvers, Paul
2016-01-01
To investigate student difficulties in solving word problems in algebra, we carried out a teaching experiment involving 51 Indonesian students (12/13 year-old) who used a digital mathematics environment. The findings were backed up by an interview study, in which eighteen students (13/14 year-old) were involved. The perspective of mathematization,…
Inhibiting Interference from Prior Knowledge: Arithmetic Intrusions in Algebra Word Problem Solving
ERIC Educational Resources Information Center
Khng, Kiat Hui; Lee, Kerry
2009-01-01
In Singapore, 6-12 year-old students are taught to solve algebra word problems with a mix of arithmetic and pre-algebraic strategies; 13-17 year-olds are typically encouraged to replace these strategies with letter-symbolic algebra. We examined whether algebra problem-solving proficiency amongst beginning learners of letter-symbolic algebra is…
The Word Problem for Solvable Lie Algebras and Groups
NASA Astrophysics Data System (ADS)
Kharlampovich, O. G.
1990-02-01
The variety of groups Z\\mathfrak{N}_2\\mathfrak{A} is given by the identity \\displaystyle \\lbrack\\lbrack x_1,\\,x_2\\rbrack,\\,\\lbrack x_3,\\,x_4\\rbrack,\\,\\lbrack x_5,\\, x_6\\rbrack,\\, x_7\\rbrack = 1,and the analogous variety of Lie algebras is given by the identity \\displaystyle (x_1x_2)(x_3x_4)(x_5x_6)x_7=0.Previously the author proved the unsolvability of the word problem for any variety of groups (respectively: Lie algebras) containing Z\\mathfrak{N}_2\\mathfrak{A}, and its solvability for any subvariety of \\mathfrak{N}_2\\mathfrak{A}. Here the word problem is investigated in varieties of Lie algebras over a field of characteristic zero and in varieties of groups contained in Z\\mathfrak{N}_2\\mathfrak{A}. It is proved that in the lattice of subvarieties of Z\\mathfrak{N}_2\\mathfrak{A} there exist arbitrary long chains in which the varieties with solvable and unsolvable word problems alternate. In particular, the variety Z\\mathfrak{N}_2\\mathfrak{A}\\cap\\mathfrak{N}_2\\mathfrak{N}_c has a solvable word problem for any c, while the variety \\mathfrak{Y}_2, given within Z\\mathfrak{N}_2\\mathfrak{A} by the identity \\displaystyle \\lbrack\\lbrack x_1,\\,\\dots,\\,x_{2c+2}\\rbrack,\\,\\lbrack y_1,\\,\\dots,\\,y_{2c+2}\\rbrack,\\lbrack z_1,\\,\\dots,\\,z_{2c}\\rbrack\\rbrack = 1,in the case of groups and by the identity \\displaystyle (x_1\\dotsb x_{2c+2})(y_1\\dotsb y_{2c+2})(z_1\\dotsb z_{2c})=0in the case of Lie algebras, has an unsolvable word problem. It is also proved that in Z\\mathfrak{N}_2\\mathfrak{A} there exists an infinite series of minimal varieties with an unsolvable word problem, i.e. varieties whose proper subvarieties all have solvable word problems.Bibliography: 17 titles.
Working Memory and Literacy as Predictors of Performance on Algebraic Word Problems
ERIC Educational Resources Information Center
Lee, Kerry; Ng, Swee-Fong; Ng, Ee-Lynn; Lim, Zee-Ying
2004-01-01
Previous studies on individual differences in mathematical abilities have shown that working memory contributes to early arithmetic performance. In this study, we extended the investigation to algebraic word problem solving. A total of 151 10-year-olds were administered algebraic word problems and measures of working memory, intelligence quotient…
Powell, Sarah R; Fuchs, Lynn S
2014-08-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2(nd)- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty.
Powell, Sarah R; Fuchs, Lynn S
2014-08-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 2(nd)- grade students, we administered (a) measures of calculations and word problems in the fall and (b) an assessment of pre-algebraic reasoning, with items that assessed solving equations and functions, in the spring. Based on the calculation and word-problem measures, we placed 148 students into 1 of 4 difficulty status categories: typically performing, calculation difficulty, word-problem difficulty, or difficulty with calculations and word problems. Analyses of variance were conducted on the 148 students; path analytic mediation analyses were conducted on the larger sample of 789 students. Across analyses, results corroborated the finding that word-problem difficulty is more strongly associated with difficulty with pre-algebraic reasoning. As an indicator of later algebra difficulty, word-problem difficulty may be a more useful predictor than calculation difficulty, and students with word-problem difficulty may require a different level of algebraic reasoning intervention than students with calculation difficulty. PMID:25309044
ERIC Educational Resources Information Center
Fuchs, Lynn S.; Zumeta, Rebecca O.; Schumacher, Robin Finelli; Powell, Sarah R.; Seethaler, Pamela M.; Hamlett, Carol L.; Fuchs, Douglas
2010-01-01
The purpose of this study was to assess the effects of schema-broadening instruction (SBI) on second graders' word-problem-solving skills and their ability to represent the structure of word problems using algebraic equations. Teachers (n = 18) were randomly assigned to conventional word-problem instruction or SBI word-problem instruction, which…
The Effect of using two variables when there are two unknowns in solving algebraic word problems
NASA Astrophysics Data System (ADS)
Mathews, Susann M.
1997-09-01
This article reports an experiment in which Algebra I students learned to translate word problems with two unknowns from the prose representation to symbolic representation using two variables (one to represent each unknown) when they first started solving word problems with two unknowns. Their performance on a test of word problems with two unknowns was compared with the results on the same test taken by students who had learned to solve word problems with two unknowns the traditional way, using only one variable to translate from prose to an algebraic equation. Four algebra teachers and 181 of their students participated in the study. A block-randomised factorial design was used. An analysis of covariance showed a statistically significant difference in the mean scores of the experimental group and the control group on this word problem test with the experimental group scoring substantially higher.
ERIC Educational Resources Information Center
Usman, Ahmed Ibrahim
2015-01-01
Knowledge and understanding of mathematical operations serves as a pre-reequisite for the successful translation of algebraic word problems. This study explored pre-service teachers' ability to recognize mathematical operations as well as use of those capabilities in constructing algebraic expressions, equations, and their solutions. The outcome…
Does Calculation or Word-Problem Instruction Provide A Stronger Route to Pre-Algebraic Knowledge?
Fuchs, Lynn S.; Powell, Sarah R.; Cirino, Paul T.; Schumacher, Robin F.; Marrin, Sarah; Hamlett, Carol L.; Fuchs, Douglas; Compton, Donald L.; Changas, Paul C.
2014-01-01
The focus of this study was connections among 3 aspects of mathematical cognition at 2nd grade: calculations, word problems, and pre-algebraic 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 pre-algebraic knowledge. Participants were 1102 children in 127 2nd-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) to pre-algebraic 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 pre-algebraic knowledge. PMID:25541565
Slower Algebra Students Meet Faster Tools: Solving Algebra Word Problems with Graphing Software
ERIC Educational Resources Information Center
Yerushalmy, Michal
2006-01-01
The article discusses the ways that less successful mathematics students used graphing software with capabilities similar to a basic graphing calculator to solve algebra problems in context. The study is based on interviewing students who learned algebra for 3 years in an environment where software tools were always present. We found differences…
ERIC Educational Resources Information Center
Ngu, Bing Hiong; Yeung, Alexander Seeshing
2012-01-01
Holyoak and Koh (1987) and Holyoak (1984) propose four critical tasks for analogical transfer to occur in problem solving. A study was conducted to test this hypothesis by comparing a multiple components (MC) approach against worked examples (WE) in helping students to solve algebra word problems in chemistry classes. The MC approach incorporated…
Effects of Graphic Organiser on Students' Achievement in Algebraic Word Problems
ERIC Educational Resources Information Center
Owolabi, Josiah; Adaramati, Tobiloba Faith
2015-01-01
This study investigated the effects of graphic organiser and gender on students' academic achievement in algebraic word problem. Three research questions and three null hypotheses were used in guiding this study. Quasi experimental research was employed and Non-equivalent pre and post test design was used. The study involved the Senior Secondary…
ERIC Educational Resources Information Center
Lee, Kerry; Ng, Ee Lynn; Ng, Swee Fong
2009-01-01
Solving algebraic word problems involves multiple cognitive phases. The authors used a multitask approach to examine the extent to which working memory and executive functioning are associated with generating problem models and producing solutions. They tested 255 11-year-olds on working memory (Counting Recall, Letter Memory, and Keep Track),…
The Effect of Using the TI-92 on Basic College Algebra Students' Ability To Solve Word Problems.
ERIC Educational Resources Information Center
Runde, Dennis C.
As part of an effort to improve community college algebra students' ability to solve word problems, a study was undertaken at Florida's Manatee Community College to determine the effects of using heuristic instruction (i.e., providing general rules for solving different types of math problems) in combination with the TI-92 calculator. The TI-92…
ERIC Educational Resources Information Center
Ngu, Bing Hiong; Yeung, Alexander Seeshing
2013-01-01
Text editing directs students' attention to the problem structure as they classify whether the texts of word problems contain sufficient, missing or irrelevant information for working out a solution. Equation worked examples emphasize the formation of a coherent problem structure to generate a solution. Its focus is on the construction of three…
Thinking and Writing Mathematically: "Achilles and the Tortoise" as an Algebraic Word Problem.
ERIC Educational Resources Information Center
Martinez, Joseph G. R.
2001-01-01
Introduces Hogben's adaptation of Zeno's paradox, "Achilles and the Tortoise", as a thinking and writing exercise. Emphasizes engaging students' imagination with creative, thought-provoking problems and involving students in evaluating their word problem-solving strategies. Describes the paradox, logical solutions, and students' mathematical…
ERIC Educational Resources Information Center
Chazan, Daniel; Sela, Hagit; Herbst, Patricio
2012-01-01
We illustrate a method, which is modeled on "breaching experiments," for studying tacit norms that govern classroom interaction around particular mathematical content. Specifically, this study explores norms that govern teachers' expectations for the doing of word problems in school algebra. Teacher study groups discussed representations of…
ERIC Educational Resources Information Center
Powell, Sarah R.; Fuchs, Lynn S.
2014-01-01
According to national mathematics standards, algebra instruction should begin at kindergarten and continue through elementary school. Most often, teachers address algebra in the elementary grades with problems related to solving equations or understanding functions. With 789 second-grade students, we administered: (1) measures of calculations and…
ERIC Educational Resources Information Center
González-Calero, José Antonio; Arnau, David; Puig, Luis; Arevalillo-Herráez, Miguel
2015-01-01
The term intensive scaffolding refers to any set of conceptual scaffolding strategies that always allow the user to find the solution to a problem. Despite the many benefits of scaffolding, some negative effects have also been reported. These are mainly related to the possibility that a student solves the problems without actually engaging in…
ERIC Educational Resources Information Center
Green, Jan
2009-01-01
In recent years, the learning of algebra by all students has become a significant national priority (Moses & Cobb, 2001; National Council of Teachers of Mathematics, 2000). Algebra is considered to be a foundational topic in mathematics (Usiskin, 1988) and some have argued that an understanding of algebra is fundamental to success in today's…
ERIC Educational Resources Information Center
Lee, Kerry; Ng, Swee Fong; Bull, Rebecca; Pe, Madeline Lee; Ho, Ringo Ho Moon
2011-01-01
Although mathematical pattern tasks are often found in elementary school curricula and are deemed a building block for algebra, a recent report (National Mathematics Advisory Panel, 2008) suggests the resources devoted to its teaching and assessment need to be rebalanced. We examined whether children's developing proficiency in solving algebraic…
Constructing a Coherent Problem Model to Facilitate Algebra Problem Solving in a Chemistry Context
ERIC Educational Resources Information Center
Ngu, Bing Hiong; Yeung, Alexander Seeshing; Phan, Huy P.
2015-01-01
An experiment using a sample of 11th graders compared text editing and worked examples approaches in learning to solve dilution and molarity algebra word problems in a chemistry context. Text editing requires students to assess the structure of a word problem by specifying whether the problem text contains sufficient, missing, or irrelevant…
ERIC Educational Resources Information Center
Fuchs, Lynn S.; Compton, Donald L.; Fuchs, Douglas; Hollenbeck, Kurstin N.; Hamlett, Carol L.; Seethaler, Pamela M.
2011-01-01
The purpose of this study was to explore the utility of a dynamic assessment (DA) of algebraic learning in predicting third graders' development of mathematics word-problem difficulty. In the fall, 122 third-grade students were assessed on a test of math word-problem skill and DA of algebraic learning. In the spring, they were assessed on…
ERIC Educational Resources Information Center
Cassidy, Jack
1991-01-01
Presents suggestions for teaching math word problems to elementary students. The strategies take into consideration differences between reading in math and reading in other areas. A problem-prediction game and four self-checking activities are included along with a magic password challenge. (SM)
ERIC Educational Resources Information Center
Arnau, David; Arevalillo-Herraez, Miguel; Puig, Luis; Gonzalez-Calero, Jose Antonio
2013-01-01
Designers of interactive learning environments with a focus on word problem solving usually have to compromise between the amount of resolution paths that a user is allowed to follow and the quality of the feedback provided. We have built an intelligent tutoring system (ITS) that is able to both track the user's actions and provide adequate…
Using Cognitive Tutor Software in Learning Linear Algebra Word Concept
ERIC Educational Resources Information Center
Yang, Kai-Ju
2015-01-01
This paper reports on a study of twelve 10th grade students using Cognitive Tutor, a math software program, to learn linear algebra word concept. The study's purpose was to examine whether students' mathematics performance as it is related to using Cognitive Tutor provided evidence to support Koedlinger's (2002) four instructional principles used…
Numerical linear algebra for reconstruction inverse problems
NASA Astrophysics Data System (ADS)
Nachaoui, Abdeljalil
2004-01-01
Our goal in this paper is to discuss various issues we have encountered in trying to find and implement efficient solvers for a boundary integral equation (BIE) formulation of an iterative method for solving a reconstruction problem. We survey some methods from numerical linear algebra, which are relevant for the solution of this class of inverse problems. We motivate the use of our constructing algorithm, discuss its implementation and mention the use of preconditioned Krylov methods.
ERIC Educational Resources Information Center
VanSciver, James H.
2009-01-01
Every assessment is a literacy test. It matters not whether the content is science, social studies, or mathematics; if students are not able to make sense of the words, their ability to decipher the meaning of the assessment questions is suspect. Comprehending the language of a task becomes even more important as educators strive to move the…
ERIC Educational Resources Information Center
Hernandez, Andrea C.
2013-01-01
This dissertation analyzes differences found in Spanish-speaking middle school and high school students in algebra-based problem solving. It identifies the accuracy differences between word problems presented in English, Spanish and numerically based problems. The study also explores accuracy differences between each subgroup of Spanish-speaking…
Constructing a coherent problem model to facilitate algebra problem solving in a chemistry context
NASA Astrophysics Data System (ADS)
Hiong Ngu, Bing; Seeshing Yeung, Alexander; Phan, Huy P.
2015-04-01
An experiment using a sample of 11th graders compared text editing and worked examples approaches in learning to solve dilution and molarity algebra word problems in a chemistry context. Text editing requires students to assess the structure of a word problem by specifying whether the problem text contains sufficient, missing, or irrelevant information for reaching a solution. Worked examples direct students to follow steps toward the solution, and its emphasis is on computation instead of the formation of a coherent problem model. Text editing yielded higher scores in a transfer test (which shared the same solution procedure as in the acquisition problems but differed in contexts), but not a similar test (which resembled acquisition problems in terms of both solution procedure and context). Results provide some theoretical support and practical implications for using text editing to develop a coherent problem model to facilitate problem-solving skills in chemistry.
Identification of Strategies Used by Fifth Graders To Solve Mathematics Word Problems.
ERIC Educational Resources Information Center
Palomares, Julio Cesar Arteaga; Hernandez, Jose Guzman
When students confront arithmetic or algebraic word problems, they develop ideas and notations during the processes of solving them by using various arithmetic strategies. Those ideas and notations are the basis for solving that type of problems. Is it possible to aid the development of students' algebraic thinking during their transition from…
Word Problems: Where Test Bias Creeps In.
ERIC Educational Resources Information Center
Chipman, Susan F.
The problem of sex bias in mathematics word problems is discussed, with references to the appropriate literature. Word problems are assessed via cognitive science analysis of word problem solving. It has been suggested that five basic semantic relations are adequate to classify nearly all story problems, namely, change, combine, compare, vary, and…
Numerical stability in problems of linear algebra.
NASA Technical Reports Server (NTRS)
Babuska, I.
1972-01-01
Mathematical problems are introduced as mappings from the space of input data to that of the desired output information. Then a numerical process is defined as a prescribed recurrence of elementary operations creating the mapping of the underlying mathematical problem. The ratio of the error committed by executing the operations of the numerical process (the roundoff errors) to the error introduced by perturbations of the input data (initial error) gives rise to the concept of lambda-stability. As examples, several processes are analyzed from this point of view, including, especially, old and new processes for solving systems of linear algebraic equations with tridiagonal matrices. In particular, it is shown how such a priori information can be utilized as, for instance, a knowledge of the row sums of the matrix. Information of this type is frequently available where the system arises in connection with the numerical solution of differential equations.
Maximum/Minimum Problems Solved Using an Algebraic Way
ERIC Educational Resources Information Center
Modica, Erasmo
2010-01-01
This article describes some problems of the maximum/minimum type, which are generally solved using calculus at secondary school, but which here are solved algebraically. We prove six algebraic properties and then apply them to this kind of problem. This didactic approach allows pupils to solve these problems even at the beginning of secondary…
ERIC Educational Resources Information Center
Lee, Kerry; Khng, Kiat Hui; Ng, Swee Fong; Ng Lan Kong, Jeremy
2013-01-01
In Singapore, primary school students are taught to use bar diagrams to represent known and unknown values in algebraic word problems. However, little is known about students' understanding of these graphical representations. We investigated whether students use and think of the bar diagrams in a concrete or a more abstract fashion. We also…
Arithmetic/Algebraic Problem-Solving and the Representation of Two Unknown Quantities
ERIC Educational Resources Information Center
Filloy, Eugenio; Rojano, Teresa; Solares, Armando
2004-01-01
We deal with the study of the senses and the meanings generated in the representation of the unknowns in the resolution of word problems involving two unknown quantities. The discussed cases show the difficulties that the students beginning the algebra learning have to deal with when using the equality between "unknown things". For them, applying…
Teaching Conceptual Model-Based Word Problem Story Grammar to Enhance Mathematics Problem Solving
ERIC Educational Resources Information Center
Xin, Yan Ping; Wiles, Ben; Lin, Yu-Ying
2008-01-01
Borrowing the concept of story grammar from reading comprehension literature, the purpose of this study was to examine the effect of teaching "word problem (WP) story grammar" on arithmetic WP solving that emphasizes the algebraic expression of mathematical relations in conceptual models. Participants were five students in Grades 4 and 5 with or…
Word Problems: A "Meme" for Our Times.
ERIC Educational Resources Information Center
Leamnson, Robert N.
1996-01-01
Discusses a novel approach to word problems that involves linear relationships between variables. Argues that working stepwise through intermediates is the way our minds actually work and therefore this should be used in solving word problems. (JRH)
Fuchs, Lynn S; Compton, Donald L; Fuchs, Douglas; Hollenbeck, Kurstin N; Hamlett, Carol L; Seethaler, Pamela M
2011-01-01
The purpose of this study was to explore the utility of a dynamic assessment (DA) of algebraic learning in predicting third graders' development of mathematics word-problem difficulty. In the fall, 122 third-grade students were assessed on a test of math word-problem skill and DA of algebraic learning. In the spring, they were assessed on word-problem performance. Logistic regression was conducted to contrast two models. One relied exclusively on the fall test of math word-problem skill to predict word-problem difficulty on the spring outcome (less than the 25th percentile). The second model relied on a combination of the fall test of math word-problem skill and the fall DA to predict the same outcome. Holding sensitivity at 87.5%, the universal screener alone resulted in a high proportion of false positives, which was practically reduced when DA was included in the prediction model. Findings are discussed in terms of a two-stage process for screening students within a responsiveness-to-intervention prevention model.
Fuchs, Lynn S.; Compton, Donald L.; Fuchs, Douglas; Hollenbeck, Kurstin N.; Hamlett, Carol L.; Seethaler, Pamela M.
2011-01-01
The purpose of this study was to explore the utility of a dynamic assessment (DA) of algebraic learning in predicting third graders’ development of mathematics word-problem difficulty. In the fall, 122 third-grade students were assessed on a test of math word-problem skill and DA of algebraic learning. In the spring, they were assessed on word-problem performance. Logistic regression was conducted to contrast two models. One relied exclusively on the fall test of math word-problem skill to predict word-problem difficulty on the spring outcome (less than the 25th percentile). The second model relied on a combination of the fall test of math word-problem skill and the fall DA to predict the same outcome. Holding sensitivity at 87.5%, the universal screener alone resulted in a high proportion of false positives, which was practically reduced when DA was included in the prediction model. Findings are discussed in terms of a two-stage process for screening students within a responsiveness-to-intervention prevention model. PMID:21685352
Clifford algebra approach to the coincidence problem for planar lattices.
Rodríguez, M A; Aragón, J L; Verde-Star, L
2005-03-01
The problem of coincidences of planar lattices is analyzed using Clifford algebra. It is shown that an arbitrary coincidence isometry can be decomposed as a product of coincidence reflections and this allows planar coincidence lattices to be characterized algebraically. The cases of square, rectangular and rhombic lattices are worked out in detail. One of the aims of this work is to show the potential usefulness of Clifford algebra in crystallography. The power of Clifford algebra for expressing geometric ideas is exploited here and the procedure presented can be generalized to higher dimensions.
Fuchs, Lynn S.; Compton, Donald L.; Fuchs, Douglas; Hollenbeck, Kurstin N.; Craddock, Caitlin F.; Hamlett, Carol L.
2008-01-01
Dynamic assessment (DA) involves helping students learn a task and indexing responsiveness to that instruction as a measure of learning potential. The purpose of this study was to explore the utility of a DA of algebraic learning in predicting 3rd graders’ development of mathematics problem solving. In the fall, 122 3rd-grade students were assessed on language, nonverbal reasoning, attentive behavior, calculations, word-problem skill, and DA. On the basis of random assignment, students received 16 weeks of validated instruction on word problems or received 16 weeks of conventional instruction on word problems. Then, students were assessed on word-problem measures proximal and distal to instruction. Structural equation measurement models showed that DA measured a distinct dimension of pretreatment ability and that proximal and distal word-problem measures were needed to account for outcome. Structural equation modeling showed that instruction (conventional vs. validated) was sufficient to account for math word-problem outcome proximal to instruction; by contrast, language, pretreatment math skill, and DA were needed to forecast learning on word-problem outcomes more distal to instruction. Findings are discussed in terms of responsiveness-to-intervention models for preventing and identifying learning disabilities. PMID:19884957
Word Frequency, Function Words and the Second Gavagai Problem
ERIC Educational Resources Information Center
Hochmann, Jean-Remy
2013-01-01
The classic gavagai problem exemplifies the difficulty to identify the referent of a novel word uttered in a foreign language. Here, we consider the reverse problem: identifying the referential part of a label. Assuming "gavagai" indicates a rabbit in a foreign language, it may very well mean ""a" rabbit" or ""that" rabbit". How can a learner know…
Powell, Sarah R.; Fuchs, Lynn S.; Cirino, Paul T.; Fuchs, Douglas; Compton, Donald L.; Changas, Paul C.
2014-01-01
The focus of the present study was enhancing word-problem and calculation achievement in ways that support pre-algebraic thinking among 2nd-grade students at risk for mathematics difficulty. Intervention relied on a multi-tier support system (i.e., responsiveness-to-intervention or RTI) in which at-risk students participate in general classroom instruction and receive supplementary small-group tutoring. Participants were 265 students in 110 classrooms in 25 schools. Teachers were randomly assigned to 3 conditions: calculation RTI, word-problem RTI, and business-as-usual control. Intervention lasted 17 weeks. Multilevel modeling indicated that calculation RTI improved calculation but not word-problem outcomes; word-problem RTI enhanced proximal word-problem outcomes as well as performance on some calculation outcomes; and word-problem RTI provided a stronger route than calculation RTI to pre-algebraic knowledge. PMID:26097244
Inverse Modelling Problems in Linear Algebra Undergraduate Courses
ERIC Educational Resources Information Center
Martinez-Luaces, Victor E.
2013-01-01
This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different…
Elementary Level Mathematics: Word Problems. Second Edition
ERIC Educational Resources Information Center
Mosrur, Ridwanul
2011-01-01
Word Problems those are also named as "Story Problems" which are well known to the students around the world. In this book there are 26 chapters which encompass diversified problems of four basic mathematical rules--addition, subtraction, multiplication and division. These problems may help students to practice more and more as well as may help…
Numerical methods on some structured matrix algebra problems
Jessup, E.R.
1996-06-01
This proposal concerned the design, analysis, and implementation of serial and parallel algorithms for certain structured matrix algebra problems. It emphasized large order problems and so focused on methods that can be implemented efficiently on distributed-memory MIMD multiprocessors. Such machines supply the computing power and extensive memory demanded by the large order problems. We proposed to examine three classes of matrix algebra problems: the symmetric and nonsymmetric eigenvalue problems (especially the tridiagonal cases) and the solution of linear systems with specially structured coefficient matrices. As all of these are of practical interest, a major goal of this work was to translate our research in linear algebra into useful tools for use by the computational scientists interested in these and related applications. Thus, in addition to software specific to the linear algebra problems, we proposed to produce a programming paradigm and library to aid in the design and implementation of programs for distributed-memory MIMD computers. We now report on our progress on each of the problems and on the programming tools.
Automatic Item Generation of Probability Word Problems
ERIC Educational Resources Information Center
Holling, Heinz; Bertling, Jonas P.; Zeuch, Nina
2009-01-01
Mathematical word problems represent a common item format for assessing student competencies. Automatic item generation (AIG) is an effective way of constructing many items with predictable difficulties, based on a set of predefined task parameters. The current study presents a framework for the automatic generation of probability word problems…
Algebraic solution of the synthesis problem for coded sequences
Leukhin, Anatolii N
2005-08-31
The algebraic solution of a 'complex' problem of synthesis of phase-coded (PC) sequences with the zero level of side lobes of the cyclic autocorrelation function (ACF) is proposed. It is shown that the solution of the synthesis problem is connected with the existence of difference sets for a given code dimension. The problem of estimating the number of possible code combinations for a given code dimension is solved. It is pointed out that the problem of synthesis of PC sequences is related to the fundamental problems of discrete mathematics and, first of all, to a number of combinatorial problems, which can be solved, as the number factorisation problem, by algebraic methods by using the theory of Galois fields and groups. (fourth seminar to the memory of d.n. klyshko)
Word Problem Solving with the Apple II.
ERIC Educational Resources Information Center
Ignatz, Mila E.
The aim of this project was to develop computer programs that will provide training in the use of a strategy for solving word problems in everyday mathematics. The strategy includes (1) classifying the problem by type, according to problem characteristics such as symbols, diagrams, relevant formulas, and arithmetic operations; (2) identifying the…
How Problem Solving Can Develop an Algebraic Perspective of Mathematics
ERIC Educational Resources Information Center
Windsor, Will
2011-01-01
SProblem solving has a long and successful history in mathematics education and is valued by many teachers as a way to engage and facilitate learning within their classrooms. The potential benefit for using problem solving in the development of algebraic thinking is that "it may broaden and develop students' mathematical thinking beyond the…
Inverse modelling problems in linear algebra undergraduate courses
NASA Astrophysics Data System (ADS)
Martinez-Luaces, Victor E.
2013-10-01
This paper will offer an analysis from a theoretical point of view of mathematical modelling, applications and inverse problems of both causation and specification types. Inverse modelling problems give the opportunity to establish connections between theory and practice and to show this fact, a simple linear algebra example in two different presentations will be discussed. Finally, several results will be presented and some conclusions proposed.
Problem Structure and Performance on Two-Step Word Problems.
ERIC Educational Resources Information Center
Hocevar, Dennis; And Others
1987-01-01
The effects of problem structure on two-step word problems are analyzed in a sample of 91 fifth through eighth graders. Results showed that sequencing the first and second steps did not cause as much trouble as understanding the first step. (RB)
Algebraic multigrid methods applied to problems in computational structural mechanics
NASA Technical Reports Server (NTRS)
Mccormick, Steve; Ruge, John
1989-01-01
The development of algebraic multigrid (AMG) methods and their application to certain problems in structural mechanics are described with emphasis on two- and three-dimensional linear elasticity equations and the 'jacket problems' (three-dimensional beam structures). Various possible extensions of AMG are also described. The basic idea of AMG is to develop the discretization sequence based on the target matrix and not the differential equation. Therefore, the matrix is analyzed for certain dependencies that permit the proper construction of coarser matrices and attendant transfer operators. In this manner, AMG appears to be adaptable to structural analysis applications.
An application of computer algebra system Cadabra to scientific problems of physics
NASA Astrophysics Data System (ADS)
Sevastianov, L. A.; Kulyabov, D. S.; Kokotchikova, M. G.
2009-12-01
In this article we present two examples solved in a new problem-oriented computer algebra system Cadabra. Solution of the same examples in widespread universal computer algebra system Maple turn out to be more difficult.
ERIC Educational Resources Information Center
Xin, Yan Ping
2008-01-01
The purpose of this study was to examine the effects of a schema-based instructional strategy that emphasizes prealgebraic conceptualization of multiplicative relations on solving arithmetic word problems with elementary students with learning disabilities or problems (LP). Introducing symbolic representation and algebraic thinking in earlier…
Generating Multiple Answers for a Word Problem with Insufficient Information
ERIC Educational Resources Information Center
Kinda, Shigehiro
2012-01-01
In mathematics learning, word problems always include sufficient information; however, in everyday situations, people sometimes encounter problems with insufficient information. Previous studies suggest that people cannot successfully handle word problems with insufficient information because they believe a word problem has only one answer and…
Linguistic Factors Affecting Correct Responses to Word Problems in Mathematics
ERIC Educational Resources Information Center
Vincent, Juliet
2009-01-01
Student underachievement on standardized math achievement tests is a major concern in American public schools. One of the speculated reasons for student underachievement is the inability to solve math word problems. Word problems are the most challenging problems in math because word problem solving requires the use of skills in language,…
Non-commutative holomorphic functions in elements of a Lie algebra and the absolute basis problem
NASA Astrophysics Data System (ADS)
Dosi, Anar A.
2009-12-01
We study the absolute basis problem in algebras of holomorphic functions in non-commuting variables generating a finite-dimensional nilpotent Lie algebra \\mathfrak{g}. This is motivated by J. L. Taylor's programme of non-commutative holomorphic functional calculus in the Lie algebra framework.
The spatial isomorphism problem for close separable nuclear C*-algebras
Christensen, Erik; Sinclair, Allan M.; Smith, Roger R.; White, Stuart A.; Winter, Wilhelm
2010-01-01
The Kadison–Kastler problem asks whether close C*-algebras on a Hilbert space must be spatially isomorphic. We establish this when one of the algebras is separable and nuclear. We also apply our methods to the study of near inclusions of C*-algebras. PMID:20080723
A new algebra core for the minimal form' problem
Purtill, M.R. . Center for Communications Research); Oliveira, J.S.; Cook, G.O. Jr. )
1991-12-20
The demands of large-scale algebraic computation have led to the development of many new algorithms for manipulating algebraic objects in computer algebra systems. For instance, parallel versions of many important algorithms have been discovered. Simultaneously, more effective symbolic representations of algebraic objects have been sought. Also, while some clever techniques have been found for improving the speed of the algebraic simplification process, little attention has been given to the issue of restructuring expressions, or transforming them into minimal forms.'' By minimal form,'' we mean that form of an expression that involves a minimum number of operations. In a companion paper, we introduce some new algorithms that are very effective at finding minimal forms of expressions. These algorithms require algebraic and combinatorial machinery that is not readily available in most algebra systems. In this paper we describe a new algebra core that begins to provide the necessary capabilities.
Fuchs, Lynn S.; Powell, Sarah R.; Seethaler, Pamela M.; Cirino, Paul T.; Fletcher, Jack M.; Fuchs, Douglas; Hamlett, Carol L.; Zumeta, Rebecca O.
2009-01-01
The purposes of this study were to assess the efficacy of remedial tutoring for 3rd graders with mathematics difficulty, to investigate whether tutoring is differentially efficacious depending on students’ math difficulty status (mathematics difficulty alone vs. mathematics plus reading difficulty), to explore transfer from number combination (NC) remediation, and to examine the transportability of the tutoring protocols. At 2 sites, 133 students were stratified on mathematics difficulty status and site and then randomly assigned to 3 conditions: control (no tutoring), tutoring on automatic retrieval of NCs (i.e., Math Flash), or tutoring on word problems with attention to the foundational skills of NCs, procedural calculations, and algebra (i.e., Pirate Math). Tutoring occurred for 16 weeks, 3 sessions per week and 20–30 min per session. Math Flash enhanced fluency with NCs with transfer to procedural computation but without transfer to algebra or word problems. Pirate Math enhanced word problem skill as well as fluency with NCs, procedural computation, and algebra. Tutoring was not differentially efficacious as a function of students’ mathematics difficulty status. The tutoring protocols proved transportable across sites. PMID:19865600
ERIC Educational Resources Information Center
McNeil, Nicole M.; Rittle-Johnson, Bethany; Hattikudur, Shanta; Petersen, Lori A.
2010-01-01
This study examined if solving arithmetic problems hinders undergraduates' accuracy on algebra problems. The hypothesis was that solving arithmetic problems would hinder accuracy because it activates an operational view of equations, even in educated adults who have years of experience with algebra. In three experiments, undergraduates (N = 184)…
Primary School Students' Strategies in Early Algebra Problem Solving Supported by an Online Game
ERIC Educational Resources Information Center
van den Heuvel-Panhuizen, Marja; Kolovou, Angeliki; Robitzsch, Alexander
2013-01-01
In this study we investigated the role of a dynamic online game on students' early algebra problem solving. In total 253 students from grades 4, 5, and 6 (10-12 years old) used the game at home to solve a sequence of early algebra problems consisting of contextual problems addressing covarying quantities. Special software monitored the…
Solving Word Problems using Schemas: A Review of the Literature.
Powell, Sarah R
2011-05-01
Solving word problems is a difficult task for students at-risk for or with learning disabilities (LD). One instructional approach that has emerged as a valid method for helping students at-risk for or with LD to become more proficient at word-problem solving is using schemas. A schema is a framework for solving a problem. With a schema, students are taught to recognize problems as falling within word-problem types and to apply a problem solution method that matches that problem type. This review highlights two schema approaches for 2(nd)- and 3(rd)-grade students at-risk for or with LD: schema-based instruction and schema-broadening instruction. A total of 12 schema studies were reviewed and synthesized. Both types of schema approaches enhanced the word-problem skill of students at-risk for or with LD. Based on the review, suggestions are provided for incorporating word-problem instruction using schemas.
Unified derivation of exact solutions to the relativistic Coulomb problem: Lie algebraic approach
NASA Astrophysics Data System (ADS)
Panahi, H.; Baradaran, M.; Savadi, A.
2015-10-01
Exact algebraic solutions of the D-dimensional Dirac and Klein-Gordon equations for the Coulomb potential are obtained in a unified treatment. It is shown that two cases are reducible to the same basic equation, which can be solved exactly. Using the Lie algebraic approach, the general exact solutions of the problem are obtained within the framework of representation theory of the sl(2) Lie algebra.
Solving Our Algebra Problem: Getting All Students through Algebra I to Improve Graduation Rates
ERIC Educational Resources Information Center
Schachter, Ron
2013-01-01
graduation as well as admission to most colleges. But taking algebra also can turn into a pathway for failure, from which some students never recover. In 2010, a national U.S. Department of Education study…
ERIC Educational Resources Information Center
Hinds, Lillian R.
Seventy Cleveland, Ohio, inner city adult illiterates, 33 from an experimental group and 37 from a contrast group, were studied to determine the efficiency and effectiveness of Words in Color or the Morphologico-Algebraic approach to teaching reading. Results indicated that the reading achievement gain of functionally illiterate adults taught by…
Problem Solving Concretely with the Word "Like"
ERIC Educational Resources Information Center
Yee, Sean
2013-01-01
While the average teenager's conversation may seem inundated with the word "like", in the mathematics classroom, teenagers use it with purpose. Linguists study the word "like" to understand and categorize comparative statements. By overlapping linguistics and mathematics education within the frame of cognitive science, this study found that high…
Reading Coaching for Math Word Problems
ERIC Educational Resources Information Center
Edwards, Sharon A.; Maloy, Robert W.; Anderson, Gordon
2009-01-01
"Math is language, too," Phyllis and David Whitin (2000) remind everyone in their informative book about reading and writing in the mathematics classroom. This means that students in elementary school math classes are learning two distinct, yet related languages--one of numbers, the other of words. These languages of numbers and words are combined…
The Association between Mathematical Word Problems and Reading Comprehension
ERIC Educational Resources Information Center
Vilenius-Tuohimaa, Piia Maria; Aunola, Kaisa; Nurmi, Jari-Erik
2008-01-01
This study aimed to investigate the interplay between mathematical word problem skills and reading comprehension. The participants were 225 children aged 9-10 (Grade 4). The children's text comprehension and mathematical word problem-solving performance was tested. Technical reading skills were investigated in order to categorise participants as…
Bilingual College Writers' Collaborative Writing of Word Problems
ERIC Educational Resources Information Center
Esquinca, Alberto
2011-01-01
Numerous researchers have studied bilingual students' performance on word problems given that reading and writing these requires that they draw on linguistic and mathematical knowledge (Barwell, 2009a, 2009b). Some researchers have studied how bilinguals write word problems in the second language, but few have considered how bilinguals use their…
An Evaluation of Interventions to Facilitate Algebra Problem Solving
ERIC Educational Resources Information Center
Mayfield, Kristin H.; Glenn, Irene M.
2008-01-01
Three participants were trained on 6 target algebra skills and subsequently received a series of 5 instructional interventions (cumulative practice, tiered feedback, feedback plus solution sequence instruction, review practice, and transfer training) in a multiple baseline across skills design. The effects of the interventions on the performance…
Excel Spreadsheets for Algebra: Improving Mental Modeling for Problem Solving
ERIC Educational Resources Information Center
Engerman, Jason; Rusek, Matthew; Clariana, Roy
2014-01-01
This experiment investigates the effectiveness of Excel spreadsheets in a high school algebra class. Students in the experiment group convincingly outperformed the control group on a post lesson assessment. The student responses, teacher observations involving Excel spreadsheet revealed that it operated as a mindtool, which formed the users'…
Application of symbolic and algebraic manipulation software in solving applied mechanics problems
NASA Technical Reports Server (NTRS)
Tsai, Wen-Lang; Kikuchi, Noboru
1993-01-01
As its name implies, symbolic and algebraic manipulation is an operational tool which not only can retain symbols throughout computations but also can express results in terms of symbols. This report starts with a history of symbolic and algebraic manipulators and a review of the literatures. With the help of selected examples, the capabilities of symbolic and algebraic manipulators are demonstrated. These applications to problems of applied mechanics are then presented. They are the application of automatic formulation to applied mechanics problems, application to a materially nonlinear problem (rigid-plastic ring compression) by finite element method (FEM) and application to plate problems by FEM. The advantages and difficulties, contributions, education, and perspectives of symbolic and algebraic manipulation are discussed. It is well known that there exist some fundamental difficulties in symbolic and algebraic manipulation, such as internal swelling and mathematical limitation. A remedy for these difficulties is proposed, and the three applications mentioned are solved successfully. For example, the closed from solution of stiffness matrix of four-node isoparametrical quadrilateral element for 2-D elasticity problem was not available before. Due to the work presented, the automatic construction of it becomes feasible. In addition, a new advantage of the application of symbolic and algebraic manipulation found is believed to be crucial in improving the efficiency of program execution in the future. This will substantially shorten the response time of a system. It is very significant for certain systems, such as missile and high speed aircraft systems, in which time plays an important role.
Effects of Numerical Surface Form in Arithmetic Word Problems
ERIC Educational Resources Information Center
Orrantia, Josetxu; Múñez, David; San Romualdo, Sara; Verschaffel, Lieven
2015-01-01
Adults' simple arithmetic performance is more efficient when operands are presented in Arabic digit (3 + 5) than in number word (three + five) formats. An explanation provided is that visual familiarity with digits is higher respect to number words. However, most studies have been limited to single-digit addition and multiplication problems. In…
The 16th Hilbert problem restricted to circular algebraic limit cycles
NASA Astrophysics Data System (ADS)
Llibre, Jaume; Ramírez, Rafael; Ramírez, Valentín; Sadovskaia, Natalia
2016-04-01
We prove the following two results. First every planar polynomial vector field of degree S with S invariant circles is Darboux integrable without limit cycles. Second a planar polynomial vector field of degree S admits at most S - 1 invariant circles which are algebraic limit cycles. In particular we solve the 16th Hilbert problem restricted to algebraic limit cycles given by circles, because a planar polynomial vector field of degree S has at most S - 1 algebraic limit cycles given by circles, and this number is reached.
The Poincaré problem, algebraic integrability and dicritical divisors
NASA Astrophysics Data System (ADS)
Galindo, C.; Monserrat, F.
We solve the Poincaré problem for plane foliations with only one dicritical divisor. Moreover, in this case, we give a simple algorithm that decides whether a foliation has a rational first integral and computes it in the affirmative case. We also provide an algorithm to compute a rational first integral of prefixed genus g≠1 of any type of plane foliation F. When the number of dicritical divisors dic(F) is larger than 2, this algorithm depends on suitable families of invariant curves. When dic(F)=2, it proves that the degree of the rational first integral can be bounded only in terms of g, the degree of F and the local analytic type of the dicritical singularities of F. The degree d of a general integral invariant curve is less than or equal to 4. Therefore, the Poincaré problem is solved in this case. There exists a valueλ∈Z>0such thatPF:=|λΔF|is a pencil and the rational mapP2⋯→P1that it defines is a rational first integral ofF. Moreover λ is the minimum of the set{α∈Z>0|dim|αΔF|⩾1}. The above clause (b) supports a very simple algorithm, our forthcoming Algorithm 2, which decides about the existence of a rational first integral of F (and computes it in the positive case) whenever dic(F)=1. Other alternative algorithms are treated in Section 4. Our remaining main results are: Assume thatFhas a rational first integral of genus g. Then, there exists a bound on the degree of the first integral depending only on the degree ofF, the genus g and the local analytic type of the dicritical singularities ofF. There exists an algorithm to decide whetherFhas a rational first integral of genus g (and to compute it, in the affirmative case) whose inputs are: g, a homogeneous 1-form definingFand the minimal resolution of the dicritical singularities ofF. Assume thatFhas a rational first integral of genus g. Then there exists a bound on the degree of the first integral which depends on the degree ofF, the genus g, the local analytic type of the
An Analysis of Children's Written Solutions to Word Problems
ERIC Educational Resources Information Center
Knifong, J. Dan; Holtan, Boyd
1976-01-01
Sixth graders wrote solutions to the word problems in the Metropolitan Achievement Test. Errors were analyzed and classified. At least 52 percent of errors were computational or clerical and could not be attributed to reading difficulties. (SD)
Kalchev, D.; Ketelsen, C.; Vassilevski, P. S.
2013-11-07
Our paper proposes an adaptive strategy for reusing a previously constructed coarse space by algebraic multigrid to construct a two-level solver for a problem with nearby characteristics. Furthermore, a main target application is the solution of the linear problems that appear throughout a sequence of Markov chain Monte Carlo simulations of subsurface flow with uncertain permeability field. We demonstrate the efficacy of the method with extensive set of numerical experiments.
Arithmetic Word-Problem-Solving in Huntington's Disease
ERIC Educational Resources Information Center
Allain, P.; Verny, C.; Aubin, G.; Pinon, K.; Bonneau, D.; Dubas, F.; Gall, D.L.
2005-01-01
The purpose of this study was to examine executive functioning in patients with Huntington's disease using an arithmetic word-problem-solving task including eight solvable problems of increasing complexity and four aberrant problems. Ten patients with Huntington's disease and 12 normal control subjects matched by age and education were tested.…
Language Repair Strategies in Bilingual Tutoring of Mathematics Word Problems
ERIC Educational Resources Information Center
Oliveira, Alandeom W.; Meskill, Carla; Judson, Darlene; Gregory, Karen; Rogers, Patterson; Imperial, Christopher J.; Casler-Failing, Shelli
2015-01-01
This study explores the "language repair strategies" (aimed at repairing communication problems) of two bilingual speakers during mathematics word problem tutoring sessions. Bilingual repair was shown to gradually shift from a linguistic to an epistemic focus during problem solving (i.e., communication became more conceptually focused…
Arithmetic Word Problem Solving: A Situation Strategy First Framework
ERIC Educational Resources Information Center
Brissiaud, Remi; Sander, Emmanuel
2010-01-01
Before instruction, children solve many arithmetic word problems with informal strategies based on the situation described in the problem. A Situation Strategy First framework is introduced that posits that initial representation of the problem activates a situation-based strategy even after instruction: only when it is not efficient for providing…
Different Procedures for Solving Mathematical Word Problems in High School
ERIC Educational Resources Information Center
Gasco, Javier; Villarroel, Jose Domingo; Zuazagoitia, Dani
2014-01-01
The teaching and learning of mathematics cannot be understood without considering the resolution of word problems. These kinds of problems not only connect mathematical concepts with language (and therefore with reality) but also promote the learning related to other scientific areas. In primary school, problems are solved by using basic…
Teaching Algebra without Algebra
ERIC Educational Resources Information Center
Kalman, Richard S.
2008-01-01
Algebra is, among other things, a shorthand way to express quantitative reasoning. This article illustrates ways for the classroom teacher to convert algebraic solutions to verbal problems into conversational solutions that can be understood by students in the lower grades. Three reasonably typical verbal problems that either appeared as or…
Kindergarten Students Solving Mathematical Word Problems
ERIC Educational Resources Information Center
Johnson, Nickey Owen
2013-01-01
The purpose of this study was to explore problem solving with kindergarten students. This line of inquiry is highly significant given that Common Core State Standards emphasize deep, conceptual understanding in mathematics as well as problem solving in kindergarten. However, there is little research on problem solving with kindergarten students.…
Direct Instruction in Math Word Problems: Students with Learning Disabilities.
ERIC Educational Resources Information Center
Wilson, Cynthia L.; Sindelar, Paul T.
1991-01-01
This study compared the effectiveness of 3 procedures for teaching 62 elementary students with learning disabilities to identify the correct algorithm in solving addition and subtraction word problems. The group receiving strategy teaching and sequencing practice problems and the group receiving strategy teaching only scored higher than…
On Development of a Problem Based Learning System for Linear Algebra with Simple Input Method
NASA Astrophysics Data System (ADS)
Yokota, Hisashi
2011-08-01
Learning how to express a matrix using a keyboard inputs requires a lot of time for most of college students. Therefore, for a problem based learning system for linear algebra to be accessible for college students, it is inevitable to develop a simple method for expressing matrices. Studying the two most widely used input methods for expressing matrices, a simpler input method for expressing matrices is obtained. Furthermore, using this input method and educator's knowledge structure as a concept map, a problem based learning system for linear algebra which is capable of assessing students' knowledge structure and skill is developed.
An efficient algorithm for the contig ordering problem under algebraic rearrangement distance.
Lu, Chin Lung
2015-11-01
Assembling a genome from short reads currently obtained by next-generation sequencing techniques often results in a collection of contigs, whose relative position and orientation along the genome being sequenced are unknown. Given two sets of contigs, the contig ordering problem is to order and orient the contigs in each set such that the genome rearrangement distance between the resulting sets of ordered and oriented contigs is minimized. In this article, we utilize the permutation groups in algebra to propose a near-linear time algorithm for solving the contig ordering problem under algebraic rearrangement distance, where the algebraic rearrangement distance between two sets of ordered and oriented contigs is the minimum weight of applicable rearrangement operations required to transform one set into the other. PMID:26247343
Overcoming the "Walls" Surrounding Word Problems
ERIC Educational Resources Information Center
Ponce, Gregorio A.; Garrison, Leslie
2004-01-01
Efforts are made to help students do better on mathematics problems without taking time from other class activities. An illustration of Mrs. Segura, a third-grade teacher at Sunflower Elementary School, is presented whose seventy percent students did not pass the fourth chapter test in mathematics.
Application of algebraic reconstruction techniques to geophysical problems
NASA Astrophysics Data System (ADS)
Peterson, J. E., Jr.
1986-04-01
Algebraic Reconstruction Techniques (ART), introduced in medical radiology, are extended in this study to seismic travel time data. The algorithms based on these techniques, developed initially for use with X-rays, must be modified for acoustic wave data. The convergence properties of these algorithms to an adequate solution and the reliability of this solution are also investigated. The algorithms developed are initially tested on synthetically derived travel time data. Travel time data from simplistic velocity models are used to determine the general behavior of the algorithms and to estimate the reliability of the reconstructed velocity field. More complex models simulate realistic velocity distributions. Results from these studies provide critical guidelines for the inversion of real travel time data. The study also investigates the amount of detail that may be determined by this method with realistic structures. Two high quality travel time data sets are inverted using ART. The experiments were carried out at the Retsof salt mine in New York and at the underground radioactive waste repository study site in Sweden (Stripa). The Stripa data set is unique in that it consists of two suites of travel time measurements; one taken while the medium was being heated by a simulated waste canister, and the other some months after the heat had been turned off. This tests the use of ART as a monitoring technique using seismic waves.
ERIC Educational Resources Information Center
Ferrara, Francesca; Sinclair, Nathalie
2016-01-01
This paper focuses on pattern generalisation as a way to introduce young students to early algebra. We build on research on patterning activities that feature, in their work with algebraic thinking, both looking for sameness recursively in a pattern (especially figural patterns, but also numerical ones) and conjecturing about function-based…
Martin, Shirley A; Bassok, Miriam
2005-04-01
Mathematical solutions to textbook word problems are correlated with semantic relations between the objects described in the problem texts. In particular, division problems usually involve functionally related objects (e.g., tulips-vases) and rarely involve categorically related objects (e.g., tulips-daisies). We examined whether middle school, high school, and college students use object relations when they solve division word problems (WP) or perform the less familiar task of representing verbal statements with algebraic equations (EQ). Both tasks involved multiplicative comparison statements with either categorically or functionally related objects (e.g., "four times as many cupcakes [commuters] as brownies [automobiles]"). Object relations affected the frequency of correct solutions in the WP task but not in the EQ task. In the latter task, object relations did affect the structure of nonalgebraic equation errors. We argue that students use object relations as "semantic cues" when they engage in the sense-making activity of mathematical modeling.
Martin, Shirley A; Bassok, Miriam
2005-04-01
Mathematical solutions to textbook word problems are correlated with semantic relations between the objects described in the problem texts. In particular, division problems usually involve functionally related objects (e.g., tulips-vases) and rarely involve categorically related objects (e.g., tulips-daisies). We examined whether middle school, high school, and college students use object relations when they solve division word problems (WP) or perform the less familiar task of representing verbal statements with algebraic equations (EQ). Both tasks involved multiplicative comparison statements with either categorically or functionally related objects (e.g., "four times as many cupcakes [commuters] as brownies [automobiles]"). Object relations affected the frequency of correct solutions in the WP task but not in the EQ task. In the latter task, object relations did affect the structure of nonalgebraic equation errors. We argue that students use object relations as "semantic cues" when they engage in the sense-making activity of mathematical modeling. PMID:16156182
Alternative Representations for Algebraic Problem Solving: When Are Graphs Better than Equations?
ERIC Educational Resources Information Center
Mielicki, Marta K.; Wiley, Jennifer
2016-01-01
Successful algebraic problem solving entails adaptability of solution methods using different representations. Prior research has suggested that students are more likely to prefer symbolic solution methods (equations) over graphical ones, even when graphical methods should be more efficient. However, this research has not tested how representation…
Muehlhoff, Rainer
2011-02-15
Existence and uniqueness of advanced and retarded fundamental solutions (Green's functions) and of global solutions to the Cauchy problem is proved for a general class of first order linear differential operators on vector bundles over globally hyperbolic Lorentzian manifolds. This is a core ingredient to CAR-/CCR-algebraic constructions of quantum field theories on curved spacetimes, particularly for higher spin field equations.
ERIC Educational Resources Information Center
Walkington, Candace; Sherman, Milan; Petrosino, Anthony
2012-01-01
This study critically examines a key justification used by educational stakeholders for placing mathematics in context--the idea that contextualization provides students with access to mathematical ideas. We present interviews of 24 ninth grade students from a low-performing urban school solving algebra story problems, some of which were…
Impact of Authenticity on Sense Making in Word Problem Solving
ERIC Educational Resources Information Center
Palm, Torulf
2008-01-01
The study presented in this paper seeks to investigate the impact of authenticity on the students' disposition to make necessary real world considerations in their word problem solving. The aim is also to gather information about the extent to which different reasons for the students' behaviors are responsible for not providing solutions that are…
A Sense-Making Approach to Word Problems
ERIC Educational Resources Information Center
Nosegbe-Okoka, Clara
2004-01-01
This study describes a conceptual teaching approach that helps students make connections between their everyday activities and mathematical word problems. It includes a brief review of the importance of teaching for understanding and describes four principles of conceptual teaching: (1) use cooperative groups; (2) allow students enough time to act…
Learning by Seeing by Doing: Arithmetic Word Problems
ERIC Educational Resources Information Center
Weber-Russell, Sylvia; LeBlanc, Mark D.
2004-01-01
"Learning by doing" in pursuit of real-world goals has received much attention from education researchers but has been unevenly supported by mathematics education software at the elementary level, particularly as it involves arithmetic word problems. In this article, we give examples of doing-oriented tools that might promote children's ability to…
Diagramming Word Problems: A Strategic Approach for Instruction
ERIC Educational Resources Information Center
van Garderen, Delinda; Scheuermann, Amy M.
2015-01-01
While often recommended as a strategy to use in order to solve word problems, drawing a diagram is a complex process that requires a good depth of understanding. Many middle school students with learning disabilities (LD) often struggle to use diagrams in an effective and efficient manner. This article presents information for teaching middle…
Is Word-Problem Solving a Form of Text Comprehension?
ERIC Educational Resources Information Center
Fuchs, Lynn S.; Fuchs, Douglas; Compton, Donald L.; Hamlett, Carol L.; Wang, Amber Y.
2015-01-01
This study's hypotheses were that (a) word-problem (WP) solving is a form of text comprehension that involves language comprehension processes, working memory, and reasoning, but (b) WP solving differs from other forms of text comprehension by requiring WP-specific language comprehension as well as general language comprehension. At the start of…
Boonen, Anton J H; de Koning, Björn B; Jolles, Jelle; van der Schoot, Menno
2016-01-01
Successfully solving mathematical word problems requires both mental representation skills and reading comprehension skills. In Realistic Math Education (RME), however, students primarily learn to apply the first of these skills (i.e., representational skills) in the context of word problem solving. Given this, it seems legitimate to assume that students from a RME curriculum experience difficulties when asked to solve semantically complex word problems. We investigated this assumption under 80 sixth grade students who were classified as successful and less successful word problem solvers based on a standardized mathematics test. To this end, students completed word problems that ask for both mental representation skills and reading comprehension skills. The results showed that even successful word problem solvers had a low performance on semantically complex word problems, despite adequate performance on semantically less complex word problems. Based on this study, we concluded that reading comprehension skills should be given a (more) prominent role during word problem solving instruction in RME.
Boonen, Anton J. H.; de Koning, Björn B.; Jolles, Jelle; van der Schoot, Menno
2016-01-01
Successfully solving mathematical word problems requires both mental representation skills and reading comprehension skills. In Realistic Math Education (RME), however, students primarily learn to apply the first of these skills (i.e., representational skills) in the context of word problem solving. Given this, it seems legitimate to assume that students from a RME curriculum experience difficulties when asked to solve semantically complex word problems. We investigated this assumption under 80 sixth grade students who were classified as successful and less successful word problem solvers based on a standardized mathematics test. To this end, students completed word problems that ask for both mental representation skills and reading comprehension skills. The results showed that even successful word problem solvers had a low performance on semantically complex word problems, despite adequate performance on semantically less complex word problems. Based on this study, we concluded that reading comprehension skills should be given a (more) prominent role during word problem solving instruction in RME. PMID:26925012
A Comparison of Two Mathematics Problem-Solving Strategies: Facilitate Algebra-Readiness
ERIC Educational Resources Information Center
Xin, Yan Ping; Zhang, Dake; Park, Joo Young; Tom, Kinsey; Whipple, Amanda; Si, Luo
2011-01-01
The authors compared a conceptual model-based problem-solving (COMPS) approach with a general heuristic instructional approach for teaching multiplication-division word-problem solving to elementary students with learning problems (LP). The results indicate that only the COMPS group significantly improved, from pretests to posttests, their…
Henson, V E
2003-02-06
The purpose of this research project was to investigate, design, and implement new algebraic multigrid (AMG) algorithms to enable the effective use of AMG in large-scale multiphysics simulation codes. These problems are extremely large; storage requirements and excessive run-time make direct solvers infeasible. The problems are highly ill-conditioned, so that existing iterative solvers either fail or converge very slowly. While existing AMG algorithms have been shown to be robust and stable for a large class of problems, there are certain problems of great interest to the Laboratory for which no effective algorithm existed prior to this research.
Arithmetic word problem solving: a Situation Strategy First framework.
Brissiaud, Rémi; Sander, Emmanuel
2010-01-01
Before instruction, children solve many arithmetic word problems with informal strategies based on the situation described in the problem. A Situation Strategy First framework is introduced that posits that initial representation of the problem activates a situation-based strategy even after instruction: only when it is not efficient for providing the numerical solution is the representation of the problem modified so that the relevant arithmetic knowledge might be used. Three experiments were conducted with Year 3 and Year 4 children. Subtraction, multiplication and division problems were created in two versions involving the same wording but different numerical values. The first version could be mentally solved with a Situation strategy (Si version) and the second with a Mental Arithmetic strategy (MA version). Results show that Si-problems are easier than MA-problems even after instruction, and, when children were asked to report their strategy by writing a number sentence, equations that directly model the situation were predominant for Si-problems but not for MA ones. Implications of the Situation Strategy First framework regarding the relation between conceptual and procedural knowledge and the development of arithmetic knowledge are discussed.
Trade-offs between grounded and abstract representations: evidence from algebra problem solving.
Koedinger, Kenneth R; Alibali, Martha W; Nathan, Mitchell J
2008-03-01
This article explores the complementary strengths and weaknesses of grounded and abstract representations in the domain of early algebra. Abstract representations, such as algebraic symbols, are concise and easy to manipulate but are distanced from any physical referents. Grounded representations, such as verbal descriptions of situations, are more concrete and familiar, and they are more similar to physical objects and everyday experience. The complementary computational characteristics of grounded and abstract representations lead to trade-offs in problem-solving performance. In prior research with high school students solving relatively simple problems, Koedinger and Nathan (2004) demonstrated performance benefits of grounded representations over abstract representations-students were better at solving simple story problems than the analogous equations. This article extends this prior work to examine both simple and more complex problems in two samples of college students. On complex problems with two references to the unknown, a "symbolic advantage" emerged, such that students were better at solving equations than analogous story problems. Furthermore, the previously observed "verbal advantage" on simple problems was replicated. We thus provide empirical support for a trade-off between grounded, verbal representations, which show advantages on simpler problems, and abstract, symbolic representations, which show advantages on more complex problems.
Algebraic Approach to the Minimum-Cost Multi-Impulse Orbit-Transfer Problem
NASA Astrophysics Data System (ADS)
Avendaño, M.; Martín-Molina, V.; Martín-Morales, J.; Ortigas-Galindo, J.
2016-08-01
We present a purely algebraic formulation (i.e. polynomial equations only) of the minimum-cost multi-impulse orbit transfer problem without time constraints, while keeping all the variables with a precise physical meaning. We apply general algebraic techniques to solve these equations (resultants, Gr\\"obner bases, etc.) in several situations of practical interest of different degrees of generality. For instance, we provide a proof of the optimality of the Hohmann transfer for the minimum fuel 2-impulse circular to circular orbit transfer problem, and we provide a general formula for the optimal 2-impulse in-plane transfer between two rotated elliptical orbits under a mild symmetry assumption on the two points where the impulses are applied (which we conjecture that can be removed).
Quantum Algorithms for Problems in Number Theory, Algebraic Geometry, and Group Theory
NASA Astrophysics Data System (ADS)
van Dam, Wim; Sasaki, Yoshitaka
2013-09-01
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the current state of quantum algorithms, focusing on algorithms for problems with an algebraic flavor that achieve an apparent superpolynomial speedup over classical computation.
ERIC Educational Resources Information Center
Xin, Yan Ping; Jitendra, Asha; Deatline-Buchman, Andria; Hickman, Wesley; Bertram, Dean
This study examined the differential effects of two instructional strategies on the acquisition, maintenance, and generalization of mathematical word problem solving by students with learning disabilities: an explicit schema-based problem solving strategy (SBI) and a traditional general heuristic instructional strategy (TI). Twenty-two middle…
How Can One Learn Mathematical Word Problems in a Second Language? A Cognitive Load Perspective
ERIC Educational Resources Information Center
Moussa-Inaty, Jase; Causapin, Mark; Groombridge, Timothy
2015-01-01
Language may ordinarily account for difficulties in solving word problems and this is particularly true if mathematical word problems are taught in a language other than one's native language. Research into cognitive load may offer a clear theoretical framework when investigating word problems because memory, specifically working memory, plays a…
Is Word-Problem Solving a Form of Text Comprehension?
Fuchs, Lynn S.; Fuchs, Douglas; Compton, Donald L.; Hamlett, Carol L.; Wang, Amber Y.
2015-01-01
This study’s hypotheses were that (a) word-problem (WP) solving is a form of text comprehension that involves language comprehension processes, working memory, and reasoning, but (b) WP solving differs from other forms of text comprehension by requiring WP-specific language comprehension as well as general language comprehension. At the start of the 2nd grade, children (n = 206; on average, 7 years, 6 months) were assessed on general language comprehension, working memory, nonlinguistic reasoning, processing speed (a control variable), and foundational skill (arithmetic for WPs; word reading for text comprehension). In spring, they were assessed on WP-specific language comprehension, WPs, and text comprehension. Path analytic mediation analysis indicated that effects of general language comprehension on text comprehension were entirely direct, whereas effects of general language comprehension on WPs were partially mediated by WP-specific language. By contrast, effects of working memory and reasoning operated in parallel ways for both outcomes. PMID:25866461
ERIC Educational Resources Information Center
Forsten, Char
2004-01-01
Children need to combine reading, thinking, and computational skills to solve math word problems. The author provides some strategies that principals can share with their teachers to help students become proficient and advanced problem-solvers. They include creating a conducive classroom environment, providing daily mental math activities, making…
ERIC Educational Resources Information Center
Duan, Xiaofang; Depaepe, Fien; Verschaffel, Lieven
2011-01-01
Word problems play a crucial role in mathematics education. However, the authenticity of word problems is quite controversial. In terms of the necessity of realistic considerations to be taken into account in the solution process, word problems have been classified into two categories: standard word problems (S-items) and problematic word problems…
Zum Problem der Neuwoerter im Russischen (On the Problem of New Words in Russian)
ERIC Educational Resources Information Center
Friederich, Wolf
1975-01-01
The article refers to Soviet neologisms and their treatment in East and West Germany. Morphological formation of new Russian words, the problem of German language equivalents and fifferences between West and East German treatment of them are dealt with. (Text is in German.) (DH)
ERIC Educational Resources Information Center
Mayer, Richard E.
In Experiments 1 and 2 subjects read a series of standard algebra story problems, and were asked to recall each problem. In Experiment 3, subjects were asked to construct problems based on certain situations (such as "train leaving stations"). Results indicated that "relational propositions" (such as "the rate in still water is 12 mph more than…
Mapping solutions to an early multiplication word problem
NASA Astrophysics Data System (ADS)
Watson, Jane M.; Mulligan, Joanne
1990-06-01
Children's solutions to a variety of multiplication and division word problems were analysed in a cross-sectional pilot study of 34 children from Grades K to 2. Responses indicated a wide range of strategies used and these were further classified into developmental levels of strategy use. Because these data reflected the SOLO Taxonomy developmental model for classifying responses, a SOLO mapping procedure was used for further analysis. In this paper, the mapping procedure is applied to only one multiplication problem strategy, repeated addition, to exemplify the procedure. The mapping device employed in the analysis isolated three components of the problem-solving procedure: the cues given by the problem, the concepts and processes used by the children, and the responses given by the children. Responses to mathematical problems from children in this age group have not previously been used to elucidate the earlier modes of functioning in the SOLO Taxonomy. In particular this paper considers the Ikonic mode and the transition into the concrete Symbolic mode.
Voila: A visual object-oriented iterative linear algebra problem solving environment
Edwards, H.C.; Hayes, L.J.
1994-12-31
Application of iterative methods to solve a large linear system of equations currently involves writing a program which calls iterative method subprograms from a large software package. These subprograms have complex interfaces which are difficult to use and even more difficult to program. A problem solving environment specifically tailored to the development and application of iterative methods is needed. This need will be fulfilled by Voila, a problem solving environment which provides a visual programming interface to object-oriented iterative linear algebra kernels. Voila will provide several quantum improvements over current iterative method problem solving environments. First, programming and applying iterative methods is considerably simplified through Voila`s visual programming interface. Second, iterative method algorithm implementations are independent of any particular sparse matrix data structure through Voila`s object-oriented kernels. Third, the compile-link-debug process is eliminated as Voila operates as an interpreter.
The Impossibility of "Real-Life" Word Problems (According to Bakhtin, Lacan, Zizek and Baudrillard)
ERIC Educational Resources Information Center
Gerofsky, Susan
2010-01-01
In recent years a great deal of work on mathematical word problems has focused on efforts to bring more of "real life" into the problems themselves and students' uptake of these problems. Following on from earlier studies of the word problem as a pedagogical and literary genre, the author argues that we cannot unproblematically assume an ability…
Bernardo, Allan B I; Calleja, Marissa O
2005-03-01
Researchers have suggested that among bilinguals, solving word problems in mathematics is influenced by linguistic factors (K. Durkin & B. Shire, 1991; L. Verschaffel, B. Greer, & E. De Corte, 2000). Others have suggested that students exhibit a strong tendency to exclude real-world constraints in solving mathematics word problems (L. Verschaffel, E. De Corte, & S. Lasure, 1994). In the present study, the authors explored the effects of stating word problems in either Filipino or English on how Filipino-English bilingual students solved word problems in which the solution required the application of real-world knowledge. The authors asked bilingual students to solve word problems in either their first or second language. For some of the word problems, real-life constraints prevented straightforward application of mathematical procedures. The authors analyzed the students' solutions to determine whether the language of the word problems affected the tendency to apply real-life constraints in the solution. Results showed that the bilingual students (a) rarely considered real-life constraints in their solutions, (b) were more successful in understanding and solving word problems that were stated in their first language, and (c) were more likely to experience failure in finding a solution to problems stated in their second language. The results are discussed in terms of the relationship between linguistic and mathematical problem-solving processes among bilinguals.
NASA Astrophysics Data System (ADS)
2015-09-01
Words matter. They are the "atoms" of written and oral communication. Students rely on words in textbooks and other instructional resources and in classroom lectures and discussions. As instructors, there are times when we need to think carefully about the words we use. Sometimes there are problems that may not be initially apparent and we may introduce confusion when we were aiming for clarity.
Profiles of Algebraic Competence
ERIC Educational Resources Information Center
Humberstone, J.; Reeve, R.A.
2008-01-01
The algebraic competence of 72 12-year-old female students was examined to identify profiles of understanding reflecting different algebraic knowledge states. Beginning algebraic competence (mapping abilities: word-to-symbol and vice versa, classifying, and solving equations) was assessed. One week later, the nature of assistance required to map…
ERIC Educational Resources Information Center
Chiu, Ming Ming
2008-01-01
The micro-time context of group processes (such as argumentation) can affect a group's micro-creativity (new ideas). Eighty high school students worked in groups of four on an algebra problem. Groups with higher mathematics grades showed greater micro-creativity, and both were linked to better problem solving outcomes. Dynamic multilevel analyses…
From Bar Diagrams to Letter-Symbolic Algebra: A Technology-Enabled Bridging
ERIC Educational Resources Information Center
Looi, C. -K.; Lim, K. -S.
2009-01-01
In the Singapore primary school Mathematics curriculum, students are taught the model method that uses bar diagrams to visualize the problem structure in a given word problem. When these students progress to secondary school, they learn the algebraic way of solving word problems. Studies (e.g. Ng et al.) have shown that poor bridging of students…
A review of vector convergence acceleration methods, with applications to linear algebra problems
NASA Astrophysics Data System (ADS)
Brezinski, C.; Redivo-Zaglia, M.
In this article, in a few pages, we will try to give an idea of convergence acceleration methods and extrapolation procedures for vector sequences, and to present some applications to linear algebra problems and to the treatment of the Gibbs phenomenon for Fourier series in order to show their effectiveness. The interested reader is referred to the literature for more details. In the bibliography, due to space limitation, we will only give the more recent items, and, for older ones, we refer to Brezinski and Redivo-Zaglia, Extrapolation methods. (Extrapolation Methods. Theory and Practice, North-Holland, 1991). This book also contains, on a magnetic support, a library (in Fortran 77 language) for convergence acceleration algorithms and extrapolation methods.
Dix, Annika; van der Meer, Elke
2015-04-01
This study investigates cognitive resource allocation dependent on fluid and numerical intelligence in arithmetic/algebraic tasks varying in difficulty. Sixty-six 11th grade students participated in a mathematical verification paradigm, while pupil dilation as a measure of resource allocation was collected. Students with high fluid intelligence solved the tasks faster and more accurately than those with average fluid intelligence, as did students with high compared to average numerical intelligence. However, fluid intelligence sped up response times only in students with average but not high numerical intelligence. Further, high fluid but not numerical intelligence led to greater task-related pupil dilation. We assume that fluid intelligence serves as a domain-general resource that helps to tackle problems for which domain-specific knowledge (numerical intelligence) is missing. The allocation of this resource can be measured by pupil dilation.
A new mathematical evaluation of smoking problem based of algebraic statistical method.
Mohammed, Maysaa J; Rakhimov, Isamiddin S; Shitan, Mahendran; Ibrahim, Rabha W; Mohammed, Nadia F
2016-01-01
Smoking problem is considered as one of the hot topics for many years. In spite of overpowering facts about the dangers, smoking is still a bad habit widely spread and socially accepted. Many people start smoking during their gymnasium period. The discovery of the dangers of smoking gave a warning sign of danger for individuals. There are different statistical methods used to analyze the dangers of smoking. In this study, we apply an algebraic statistical method to analyze and classify real data using Markov basis for the independent model on the contingency table. Results show that the Markov basis based classification is able to distinguish different date elements. Moreover, we check our proposed method via information theory by utilizing the Shannon formula to illustrate which one of these alternative tables is the best in term of independent.
A Graphical Solution of Certain Selected Problems
ERIC Educational Resources Information Center
Hess, Lindsay L.; Hess, Adrien L.
1978-01-01
Graphical solutions are illustrated for several algebra problems including finding roots of a quadratic equation, solving mixture and motion word problems, factoring the difference of two squares, and constructing the square root of a positive number. (MN)
Endogenous control and task representation: an fMRI study in algebraic problem-solving.
Stocco, Andrea; Anderson, John R
2008-07-01
The roles of prefrontal and anterior cingulate cortices have been widely studied, yet little is known on how they interact to enable complex cognitive abilities. We investigated this issue in a complex yet well-defined symbolic paradigm: algebraic problem solving. In our experimental problems, the demands for retrieving arithmetic facts and maintaining intermediate problem representations were manipulated separately. An analysis of functional brain images acquired while participants were solving the problems confirmed that prefrontal regions were affected by the retrieval of arithmetic facts, but only scarcely by the need to manipulate intermediate forms of the equations, hinting at a specific role in memory retrieval. Hemodynamic activity in the dorsal cingulate, on the contrary, increased monotonically as more information processing steps had to be taken, independent of their nature. This pattern was essentially mimicked in the caudate nucleus, suggesting a related functional role in the control of cognitive actions. We also implemented a computational model within the Adaptive Control of Thought-Rational (ACT-R) cognitive architecture, which was able to reproduce both the behavioral data and the time course of the hemodynamic activity in a number of relevant regions of interest. Therefore, imaging results and computer simulation provide evidence that symbolic cognition can be explained by the functional interaction of medial structures supporting control and serial execution, and prefrontal cortices engaged in the on-line retrieval of specific relevant information. PMID:18284348
Young Filipino Students Making Sense of Arithmetic Word Problems in English
ERIC Educational Resources Information Center
Bautista, Debbie; Mulligan, Joanne; Mitchelmore, Michael
2009-01-01
Young Filipino children are expected to solve mathematical word problems in English, a task which they typically encounter only in schools. In this exploratory study, task-based interviews were conducted with seven Filipino children from a public school. The children were asked to read and solve addition and subtraction word problems in English or…
Why Do Disadvantaged Filipino Children Find Word Problems in English Difficult?
ERIC Educational Resources Information Center
Bautista, Debbie; Mulligan, Joanne
2010-01-01
Young Filipino students are expected to solve mathematical word problems in English, a language that many encounter only in schools. Using individual interviews of 17 Filipino children, we investigated why word problems in English are difficult and the extent to which the language interferes with performance. Results indicate that children could…
Duality of Mathematical Thinking When Making Sense of Simple Word Problems: Theoretical Essay
ERIC Educational Resources Information Center
Polotskaia, Elena; Savard, Annie; Freiman, Viktor
2015-01-01
This essay proposes a reflection on the learning difficulties and teaching approaches associated with arithmetic word problem solving. We question the development of word problem solving skills in the early grades of elementary school. We are trying to revive the discussion because first, the knowledge in question--reversibility of arithmetic…
Text Integration and Mathematical Connections: A Computer Model of Arithmetic Word Problem Solving.
ERIC Educational Resources Information Center
LeBlanc, Mark D.; Weber-Russell, Sylvia
1996-01-01
A growing body of empirical and theoretical work indicates that young children (grades K-3) have difficulties solving word problems because of deficient language and text comprehension strategies. Describes a computer simulation designed to model working memory demands in "bottom-up" comprehension of arithmetic word problems, offering a…
The Motivation of Secondary School Students in Mathematical Word Problem Solving
ERIC Educational Resources Information Center
Gasco, Javier; Villarroel, Jose-Domingo
2014-01-01
Introduction: Motivation is an important factor in the learning of mathematics. Within this area of education, word problem solving is central in most mathematics curricula of Secondary School. The objective of this research is to detect the differences in motivation in terms of the strategies used to solve word problems. Method: It analyzed the…
Helping Students with Emotional and Behavioral Disorders Solve Mathematics Word Problems
ERIC Educational Resources Information Center
Alter, Peter
2012-01-01
The author presents a strategy for helping students with emotional and behavioral disorders become more proficient at solving math word problems. Math word problems require students to go beyond simple computation in mathematics (e.g., adding, subtracting, multiplying, and dividing) and use higher level reasoning that includes recognizing relevant…
Does Calculation or Word-Problem Instruction Provide a Stronger Route to Prealgebraic Knowledge?
ERIC Educational Resources Information Center
Fuchs, Lynn S.; Powell, Sarah R.; Cirino, Paul T.; Schumacher, Robin F.; Marrin, Sarah; Hamlett, Carol L.; Fuchs, Douglas; Compton, Donald L.; Changas, Paul C.
2014-01-01
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…
Does Understanding Relational Terminology Mediate Effects of Intervention on Compare Word Problems?
ERIC Educational Resources Information Center
Schumacher, Robin F.; Fuchs, Lynn S.
2012-01-01
The purpose of this study was to assess whether understanding relational terminology (i.e., "more, less," and "fewer") mediates the effects of intervention on compare word problems. Second-grade classrooms (N = 31) were randomly assigned to one of three conditions: researcher-designed word-problem intervention, researcher-designed calculation…
An Exploratory Study Contrasting High- and Low-Achieving Students' Percent Word Problem Solving
ERIC Educational Resources Information Center
Jitendra, Asha K.; Star, Jon R.
2012-01-01
This study evaluated whether schema-based instruction (SBI), a promising method for teaching students to represent and solve mathematical word problems, impacted the learning of percent word problems. Of particular interest was the extent that SBI improved high- and low-achieving students' learning and to a lesser degree on the indirect effect of…
ERIC Educational Resources Information Center
Bae, Young Seh; Chiang, Hsu-Min; Hickson, Linda
2015-01-01
This study examined the difference between children with autism spectrum disorders (ASD) and children with typical development (TD) in mathematical word problem solving ability and the factors associated with these children's word problem-solving ability. A total of 20 children with ASD and 20 children with TD participated in this study.…
Assessing the Effect of Language Demand in Bundles of Math Word Problems
ERIC Educational Resources Information Center
Banks, Kathleen; Jeddeeni, Ahmad; Walker, Cindy M.
2016-01-01
Differential bundle functioning (DBF) analyses were conducted to determine whether seventh and eighth grade second language learners (SLLs) had lower probabilities of answering bundles of math word problems correctly that had heavy language demands, when compared to non-SLLs of equal math proficiency. Math word problems on each of four test forms…
ERIC Educational Resources Information Center
Beitzel, Brian D.; Staley, Richard K.; DuBois, Nelson F.
2011-01-01
Previous research has cast doubt on the efficacy of utilizing external representations as an aid to solving word problems. The present study replicates previous findings that concrete representations hinder college students' ability to solve probability word problems, and extends those findings to apply to a multimedia instructional context. Our…
Cognitive Strategy Instruction for Teaching Word Problems to Primary-Level Struggling Students
ERIC Educational Resources Information Center
Pfannenstiel, Kathleen Hughes; Bryant, Diane Pedrotty; Bryant, Brian R.; Porterfield, Jennifer A.
2015-01-01
Students with mathematics difficulties and learning disabilities (LD) typically struggle with solving word problems. These students often lack knowledge about efficient, cognitive strategies to utilize when solving word problems. Cognitive strategy instruction has been shown to be effective in teaching struggling students how to solve word…
Using Number Lines to Solve Math Word Problems: A Strategy for Students with Learning Disabilities
ERIC Educational Resources Information Center
Gonsalves, Nicola; Krawec, Jennifer
2014-01-01
Students with learning disabilities (LD) consistently struggle with word problem solving in mathematics classes. This difficulty has made curricular, state, and national tests particularly stressful, as word problem solving has become a predominant feature of such student performance assessments. Research suggests that students with LD perform…
ERIC Educational Resources Information Center
Bae, Young Seh
2013-01-01
Mathematical Word Problem Solving of Students with Autistic Spectrum Disorders and Students with Typical Development Young Seh Bae This study investigated mathematical word problem solving and the factors associated with the solution paths adopted by two groups of participants (N=40), students with autism spectrum disorders (ASDs) and typically…
Trade-Offs between Grounded and Abstract Representations: Evidence from Algebra Problem Solving
ERIC Educational Resources Information Center
Koedinger, Kenneth R.; Alibali, Martha W.; Nathan, Mitchell J.
2008-01-01
This article explores the complementary strengths and weaknesses of grounded and abstract representations in the domain of early algebra. Abstract representations, such as algebraic symbols, are concise and easy to manipulate but are distanced from any physical referents. Grounded representations, such as verbal descriptions of situations, are…
ERIC Educational Resources Information Center
Booth, Julie L.; Lange, Karin E.; Koedinger, Kenneth R.; Newton, Kristie J.
2013-01-01
In a series of two in vivo experiments, we examine whether correct and incorrect examples with prompts for self-explanation can be effective for improving students' conceptual understanding and procedural skill in Algebra when combined with guided practice. In Experiment 1, students working with the Algebra I Cognitive Tutor were randomly assigned…
ERIC Educational Resources Information Center
Booth, Julie L.; Lange, Karin E.; Koedinger, Kenneth R.; Newton, Kristie J.
2013-01-01
In a series of two "in vivo" experiments, we examine whether correct and incorrect examples with prompts for self-explanation can be effective for improving students' conceptual understanding and procedural skill in Algebra when combined with guided practice. In Experiment 1, students working with the Algebra I Cognitive Tutor were randomly…
Application of Schema Theory to the Instruction of Arithmetic Word Problem Solving Skills.
ERIC Educational Resources Information Center
Tsai, Chia-jer; Derry, Sharon J.
An understanding-based approach to teaching arithmetic word problems is used in the Training Arithmetic Problem Solving Skills (TAPS) research project, for which four semantic schemas or problem representations have been revised and adopted: Combine, Compare, Change, and Vary. It is hypothesized that a good problem solver identifies the schema of…
ERIC Educational Resources Information Center
Lubin, Amélie; Vidal, Julie; Lanoë, Céline; Houdé, Olivier; Borst, Grégoire
2013-01-01
Solving simple arithmetic word problems is a major ability that children must acquire throughout the primary-grade mathematics curriculum. However, this skill is often challenging for them. For instance, "unknown referent problems" are more difficult to solve than "unknown compare problems." In unknown compare problems, the…
Acceleration of multiple solution of a boundary value problem involving a linear algebraic system
NASA Astrophysics Data System (ADS)
Gazizov, Talgat R.; Kuksenko, Sergey P.; Surovtsev, Roman S.
2016-06-01
Multiple solution of a boundary value problem that involves a linear algebraic system is considered. New approach to acceleration of the solution is proposed. The approach uses the structure of the linear system matrix. Particularly, location of entries in the right columns and low rows of the matrix, which undergo variation due to the computing in the range of parameters, is used to apply block LU decomposition. Application of the approach is considered on the example of multiple computing of the capacitance matrix by method of moments used in numerical electromagnetics. Expressions for analytic estimation of the acceleration are presented. Results of the numerical experiments for solution of 100 linear systems with matrix orders of 1000, 2000, 3000 and different relations of variated and constant entries of the matrix show that block LU decomposition can be effective for multiple solution of linear systems. The speed up compared to pointwise LU factorization increases (up to 15) for larger number and order of considered systems with lower number of variated entries.
ERIC Educational Resources Information Center
Rosales, Javier; Vicente, Santiago; Chamoso, Jose M.; Munez, David; Orrantia, Josetxu
2012-01-01
Word problem solving involves the construction of two different mental representations, namely, mathematical and situational. Although educational research in word problem solving has documented different kinds of instruction at these levels, less is known about how both representational levels are evoked during word problem solving in day-to-day…
ERIC Educational Resources Information Center
Cheng, Lu Pien
2015-01-01
In this study, ways in which 9-year old students from one Singapore school solved 1-step and 2-step word problems based on the three semantic structures were examined. The students' work and diagrams provided insights into the range of errors in word problem solving for 1- step and 2-step word problems. In particular, the errors provided some…
ERIC Educational Resources Information Center
Van Dooren, Wim; De Bock, Dirk; Vleugels, Kim; Verschaffel, Lieven
2010-01-01
Upper primary school children often routinely apply proportional methods to missing-value problems, even when it is inappropriate. We tested whether this tendency could be weakened if children were not required to produce computational answers to such problems. A total of 75 sixth graders were asked to classify 9 word problems of three types (3…
ERIC Educational Resources Information Center
Parmar, Rene S.; Cawley, John F.
1994-01-01
Matrix organization can be used to construct math word problems for children with mild disabilities. Matrix organization specifies the characteristics of problems, such as problem theme or setting, operations, level of computation complexity, reading vocabulary level, and need for classification. A sample scope and sequence and 16 sample word…
The Effectiveness of Using the Model Method to Solve Word Problems
ERIC Educational Resources Information Center
Bao, Lei
2016-01-01
The aim of this study is to investigate whether the model method is effective to assist primary students to solve word problems. The model method not only provides students with an opportunity to interpret the problem by drawing the rectangular bar but also helps students to visually represent problem situations and relevant relationships on the…
The Efficacy of Using Diagrams When Solving Probability Word Problems in College
ERIC Educational Resources Information Center
Beitzel, Brian D.; Staley, Richard K.
2015-01-01
Previous experiments have shown a deleterious effect of visual representations on college students' ability to solve total- and joint-probability word problems. The present experiments used conditional-probability problems, known to be more difficult than total- and joint-probability problems. The diagram group was instructed in how to use…
Does understanding relational terminology mediate effects of intervention on compare word problems?
Schumacher, Robin F; Fuchs, Lynn S
2012-04-01
The purpose of this study was to assess whether understanding relational terminology (i.e., more, less, and fewer) mediates the effects of intervention on compare word problems. Second-grade classrooms (N=31) were randomly assigned to one of three conditions: researcher-designed word-problem intervention, researcher-designed calculation intervention, or business-as-usual (teacher-designed) control. Students in word-problem intervention classrooms received instruction on the compare problem type, which included a focus on understanding relational terminology within compare word problems. Analyses, which accounted for variance associated with classroom clustering, indicated that (a) compared with the calculation intervention and business-as-usual conditions, word-problem intervention significantly increased performance on all three subtypes of compare problems and on understanding relational terminology, and (b) the intervention effect was fully mediated by students' understanding of relational terminology for one subtype of compare problems and partially mediated by students' understanding of relational terminology for the other two subtypes.
Process Inquiry: Analysis of Oral Problem-Solving Skills in Mathematics of Engineering Students
ERIC Educational Resources Information Center
Trance, Naci John C.
2013-01-01
This paper presents another effort in determining the difficulty of engineering students in terms of solving word problems. Students were presented with word problems in algebra. Then, they were asked to solve the word problems orally; that is, before they presented their written solutions, they were required to explain how they understood the…
NASA Technical Reports Server (NTRS)
Iachello, Franco
1995-01-01
An algebraic formulation of quantum mechanics is presented. In this formulation, operators of interest are expanded onto elements of an algebra, G. For bound state problems in nu dimensions the algebra G is taken to be U(nu + 1). Applications to the structure of molecules are presented.
Algebra and Problem-Solving in Down Syndrome: A Study with 15 Teenagers
ERIC Educational Resources Information Center
Martinez, Elisabetta Monari; Pellegrini, Katia
2010-01-01
There is a common opinion that mathematics is difficult for persons with Down syndrome, because of a weakness in numeracy and in abstract thinking. Since 1996, some single case studies have suggested that new opportunities in mathematics are possible for these students: some of them learned algebra and also learned to use equations in…
Strategies Used by Second-Year Algebra Students to Solve Problems
ERIC Educational Resources Information Center
Senk, Sharon L.; Thompson, Denisse R.
2006-01-01
This Brief Report describes a secondary analysis of the solutions written by 306 second-year algebra students to four constructed-response items representative of content at this level. The type of solution (symbolic, graphical, or numerical) used most frequently varied by item. Curriculum effects were observed. Students studying from the second…
Reading-Enhanced Word Problem Solving: A Theoretical Model
ERIC Educational Resources Information Center
Capraro, Robert M.; Capraro, Mary Margaret; Rupley, William H.
2012-01-01
There is a reciprocal relationship between mathematics and reading cognition. Metacognitive training within reading-enhanced problem solving should facilitate students developing an awareness of what good readers do when reading for meaning in solving mathematical problems enabling them to apply these strategies. The constructs for each cognitive…
A Strategy for Improving US Middle School Student Mathematics Word Problem Solving Performance
NASA Technical Reports Server (NTRS)
Thomas, Valerie L.
2004-01-01
U.S. middle school students have difficulty understanding and solving mathematics word problems. Their mathematics performance on the Third International Mathematics and Science Study (TIMMS) is far below their international peers, and minority students are less likely than high socioeconomic status (SES) White/Asian students to be exposed to higher-level mathematics concepts. Research literature also indicates that when students use both In-School and Out-of-School knowledge and experiences to create authentic mathematics word problems, student achievement improves. This researcher developed a Strategy for improving mathematics problem solving performance and a Professional Development Model (PDM) to effectively implement the Strategy.
ERIC Educational Resources Information Center
Gunbas, Nilgun
2012-01-01
The purpose of this study was to investigate the effect of a computer-based story on sixth grade students' mathematics word problem solving achievement. Problems were embedded in a story presented on a computer, and then compared to a paper-based story and to a condition that presented the problems as typical, isolated words problems. One hundred…
Factors Influencing Filipino Children's Solutions to Addition and Subtraction Word Problems
ERIC Educational Resources Information Center
Bautista, Debbie; Mitchelmore, Michael; Mulligan, Joanne
2009-01-01
Young Filipino children are expected to solve mathematical word problems in English, which is not their mother tongue. Because of this, it is often assumed that Filipino children have difficulties in solving problems because they cannot read or comprehend what they have read. This study tested this assumption by determining whether presenting word…
ERIC Educational Resources Information Center
Taber, Mary R.
2013-01-01
Mathematics can be a difficult topic both to teach and to learn. Word problems specifically can be difficult for students with disabilities because they have to conceptualize what the problem is asking for, and they must perform the correct operation accurately. Current trends in mathematics instruction stem from the National Council of Teachers…
Students' Mathematics Word Problem-Solving Achievement in a Computer-Based Story
ERIC Educational Resources Information Center
Gunbas, N.
2015-01-01
The purpose of this study was to investigate the effect of a computer-based story, which was designed in anchored instruction framework, on sixth-grade students' mathematics word problem-solving achievement. Problems were embedded in a story presented on a computer as computer story, and then compared with the paper-based version of the same…
Application of Graph Theory in an Intelligent Tutoring System for Solving Mathematical Word Problems
ERIC Educational Resources Information Center
Nabiyev, Vasif V.; Çakiroglu, Ünal; Karal, Hasan; Erümit, Ali K.; Çebi, Ayça
2016-01-01
This study is aimed to construct a model to transform word "motion problems" in to an algorithmic form in order to be processed by an intelligent tutoring system (ITS). First; categorizing the characteristics of motion problems, second; suggesting a model for the categories were carried out. In order to solve all categories of the…
ERIC Educational Resources Information Center
Munez, David; Orrantia, Josetxu; Rosales, Javier
2013-01-01
This study explored the effectiveness of external representations presented together with compare word problems, and whether such effectiveness was moderated by working memory. Participants were 49 secondary school students. Each participant solved 48 problems presented in 4 presentation types that included 2 difficulty treatments (number of steps…
Learning To Solve Word Problems in a Middle School Vision Class.
ERIC Educational Resources Information Center
Krebs, Cathryn S.
2001-01-01
A resource vision teacher describes activities to develop skills in solving mathematical word problems by three seventh graders with severe visual impairments. Students kept portfolios of problems they actually experienced in their daily lives. Success was achieved through providing an optimal environment, active involvement, self-assessment, and…
A Comparison of Updating Processes in Children Good or Poor in Arithmetic Word Problem-Solving
ERIC Educational Resources Information Center
Passolunghi, Maria Chiara; Pazzaglia, Francesca
2005-01-01
This study examines the updating ability of poor or good problem solvers. Seventy-eight fourth-graders, 43 good and 35 poor arithmetic word problem-solvers, performed the Updating Test used in Palladino et al. [Palladino, P., Cornoldi, C., De Beni, R., and Pazzaglia F. (2002). Working memory and updating processes in reading comprehension. Memory…
Manipulatives and Number Sentences in Computer-Aided Arithmetic Word Problem Solving.
ERIC Educational Resources Information Center
Stellingwerf, Berend P.; van Lieshout, Ernest C. D. M.
1999-01-01
Investigates the relative contribution of two main components often used in the instruction of arithmetic and word-problem solving to first-grade children and children with learning problems: external representation with manipulatives and formal mathematical representation with number sequences. Four computer-aided treatments were developed along…
Solving Word Problems about Time: The Effects of Speed and Space Information.
ERIC Educational Resources Information Center
Senechal, Monique
This study investigated how preadolescents and adolescents solve problems involving three temporal dimensions. Specifically examined was the question of whether speed and space information would influence the time judgments of 90 subjects 9, 12, and 15 years of age who solved 16 word problems describing the displacements of two cars. The problems…
Scaffold Seeking: A Reverse Design of Scaffolding in Computer-Supported Word Problem Solving
ERIC Educational Resources Information Center
Cheng, Hercy N. H.; Yang, Euphony F. Y.; Liao, Calvin C. Y.; Chang, Ben; Huang, Yana C. Y.; Chan, Tak-Wai
2015-01-01
Although well-designed scaffolding may assist students to accomplish learning tasks, its insufficient capability to dynamically assess students' abilities and to adaptively support them may result in the problem of overscaffolding. Our previous project has also shown that students using scaffolds to solve mathematical word problems for a long time…
ERIC Educational Resources Information Center
Kempert, Sebastian; Saalbach, Henrik; Hardy, Ilonca
2011-01-01
Previous research has emphasized the importance of language for learning mathematics. This is especially true when mathematical problems have to be extracted from a meaningful context, as in arithmetic word problems. Bilingual learners with a low command of the instructional language thus may face challenges when dealing with mathematical…
Zhuk, Sergiy
2013-10-15
In this paper we present Kalman duality principle for a class of linear Differential-Algebraic Equations (DAE) with arbitrary index and time-varying coefficients. We apply it to an ill-posed minimax control problem with DAE constraint and derive a corresponding dual control problem. It turns out that the dual problem is ill-posed as well and so classical optimality conditions are not applicable in the general case. We construct a minimizing sequence u-circumflex{sub {epsilon}} for the dual problem applying Tikhonov method. Finally we represent u-circumflex{sub {epsilon}} in the feedback form using Riccati equation on a subspace which corresponds to the differential part of the DAE.
ERIC Educational Resources Information Center
Swanson, H. Lee; Lussier, Cathy; Orosco, Michael
2013-01-01
This study investigated the role of strategy instruction and cognitive abilities on word problem solving accuracy in children with math difficulties (MD). Elementary school children (N = 120) with and without MD were randomly assigned to 1 of 4 conditions: general-heuristic (e.g., underline question sentence), visual-schematic presentation…
Numerical algebraic geometry and algebraic kinematics
NASA Astrophysics Data System (ADS)
Wampler, Charles W.; Sommese, Andrew J.
In this article, the basic constructs of algebraic kinematics (links, joints, and mechanism spaces) are introduced. This provides a common schema for many kinds of problems that are of interest in kinematic studies. Once the problems are cast in this algebraic framework, they can be attacked by tools from algebraic geometry. In particular, we review the techniques of numerical algebraic geometry, which are primarily based on homotopy methods. We include a review of the main developments of recent years and outline some of the frontiers where further research is occurring. While numerical algebraic geometry applies broadly to any system of polynomial equations, algebraic kinematics provides a body of interesting examples for testing algorithms and for inspiring new avenues of work.
Language, arithmetic word problems, and deaf students: Linguistic strategies used to solve tasks
NASA Astrophysics Data System (ADS)
Zevenbergen, Robyn; Hyde, Merv; Power, Des
2001-12-01
There has been limited examination of the intersection between language and arithmetic in the performance of deaf students, although some previous research has shown that deaf and hearing-impaired1 students are delayed in both their language acquisition and arithmetic performance. This paper examines the performance of deaf and hearing-impaired students in South-East Queensland, Australia, in solving arithmetic word problems. It was found that the subjects' solutions of word problems confirmed trends for hearing students, but that their performance was delayed in comparison. The results confirm other studies where deaf and hearing-impaired students are delayed in their language acquisition and this impacts on their capacity to successfully undertake the resolution of word problems.
Tense and aspect in word problems about motion: diagram, gesture, and the felt experience of time
NASA Astrophysics Data System (ADS)
de Freitas, Elizabeth; Zolkower, Betina
2015-09-01
Word problems about motion contain various conjugated verb forms. As students and teachers grapple with such word problems, they jointly operationalize diagrams, gestures, and language. Drawing on findings from a 3-year research project examining the social semiotics of classroom interaction, we show how teachers and students use gesture and diagram to make sense of complex verb forms in such word problems. We focus on the grammatical category of "aspect" for how it broadens the concept of verb tense. Aspect conveys duration and completion or frequency of an event. The aspect of a verb defines its temporal flow (or lack thereof) and the location of a vantage point for making sense of this durational process.
Improving Student Achievement in Solving Mathematical Word Problems.
ERIC Educational Resources Information Center
Roti, Joan; Trahey, Carol; Zerafa, Susan
This report describes a program for improving students' comprehension of the language of mathematical problems. The targeted population consists of 5th and 6th grade multi-age students and multi-age learners with special needs at a middle school located outside a major city in a Midwestern community. Evidence for the existence of this problem…
A Working Memory Model Applied to Mathematical Word Problem Solving
ERIC Educational Resources Information Center
Alamolhodaei, Hassan
2009-01-01
The main objective of this study is (a) to explore the relationship among cognitive style (field dependence/independence), working memory, and mathematics anxiety and (b) to examine their effects on students' mathematics problem solving. A sample of 161 school girls (13-14 years old) were tested on (1) the Witkin's cognitive style (Group Embedded…
A Kind Word for Bullshit: The Problem of Academic Writing
ERIC Educational Resources Information Center
Eubanks, Philip; Schaeffer, John D.
2008-01-01
The phrase "academic bullshit" presents compositionists with a special dilemma. Because compositionists study, teach, and produce academic writing, they are open to the accusation that they both tolerate and perpetuate academic bullshit. We argue that confronting this problem must begin with a careful definition of "bullshit" and "academic…
ERIC Educational Resources Information Center
Wong, Wing-Kwong; Hsu, Sheng-Cheng; Wu, Shih-Hung; Lee, Cheng-Wei; Hsu, Wen-Lian
2007-01-01
Computer-assisted instruction systems have been broadly applied to help students solve math word problem. The majority of such systems, which are based on an instructor-initiating instruction strategy, provide pre-designed problems for the learners. When learners are asked to solve a word problem, the system will instruct the learners what to do.…
ERIC Educational Resources Information Center
Peake, Christian; Jiménez, Juan E.; Rodríguez, Cristina; Bisschop, Elaine; Villarroel, Rebeca
2015-01-01
Arithmetic word problem (AWP) solving is a highly demanding task for children with learning disabilities (LD) since verbal and mathematical information have to be integrated. This study examines specifically how syntactic awareness (SA), the ability to manage the grammatical structures of language, affects AWP solving. Three groups of children in…
Semantic Similarity Graphs of Mathematics Word Problems: Can Terminology Detection Help?
ERIC Educational Resources Information Center
John, Rogers Jeffrey Leo; Passonneau, Rebecca J.; McTavish, Thomas S.
2015-01-01
Curricula often lack metadata to characterize the relatedness of concepts. To investigate automatic methods for generating relatedness metadata for a mathematics curriculum, we first address the task of identifying which terms in the vocabulary from mathematics word problems are associated with the curriculum. High chance-adjusted interannotator…
Learning to Solve Addition and Subtraction Word Problems in English as an Imported Language
ERIC Educational Resources Information Center
Verzosa, Debbie Bautista; Mulligan, Joanne
2013-01-01
This paper reports an intervention phase of a design study aimed to assist second-grade Filipino children in solving addition word problems in English, a language they primarily encounter only in school. With Filipino as the medium of instruction, an out-of-school pedagogical intervention providing linguistic and representational scaffolds was…
An Analysis of Word Problems in School Mathematics Texts: Operation of Addition and Subtraction
ERIC Educational Resources Information Center
Singh, Parmjit
2006-01-01
This paper discusses the types of word problems represented in Malaysia's primary one, primary two and primary three mathematics texts based on Van De Walle's model (1998) in the operations of addition and subtraction. A test was constructed to measure students' success based on this model. The data from this study indicates that the Malaysian…
Do Curriculum-Based Measures Predict Performance on Word-Problem-Solving Measures?
ERIC Educational Resources Information Center
Sisco-Taylor, Dennis; Fung, Wenson; Swanson, H. Lee
2015-01-01
This study examined whether curriculum-based measures (CBMs) of math word-problem contributed unique variance in predictions of performance on high-stakes tests, beyond the contribution of calculation and reading skills. CBMs were administered to a representative sample of 142 third-grade students at three time points. Results indicate that…
ERIC Educational Resources Information Center
Adiguzel, Tufan; Akpinar, Yavuz
2004-01-01
Instructional resources that employ multiple representations have become commonplace in mathematics classrooms. This study will present computer software, LaborScale which was designed to improve seventh grade students' word problem-solving skills through computer-based multiple representations including graphic, symbolic, and audio…
ERIC Educational Resources Information Center
Zheng, Xinhua; Flynn, Lindsay J.; Swanson, H. Lee
2013-01-01
This article provides a quantitative synthesis of the published literature on word problem solving intervention studies for children with math disabilities (MD). Seven group and eight single-subject design studies met inclusion criteria. Mean effect sizes ("ES"s) for solution accuracy for group design studies were 0.95 (SE = 0.19) for children…
ERIC Educational Resources Information Center
Björn, Piia Maria; Aunola, Kaisa; Nurmi, Jari-Erik
2016-01-01
This longitudinal study aimed to investigate the extent to which primary school text comprehension predicts mathematical word problem-solving skills in secondary school among Finnish students. The participants were 224 fourth graders (9-10 years old at the baseline). The children's text-reading fluency, text comprehension and basic calculation…
ERIC Educational Resources Information Center
Swanson, H. Lee; Lussier, Catherine M.; Orosco, Michael J.
2015-01-01
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…
The Effects of Dynamic Strategic Math on English Language Learners' Word Problem Solving
ERIC Educational Resources Information Center
Orosco, Michael J.; Swanson, H. Lee; O'Connor, Rollanda; Lussier, Cathy
2013-01-01
English language learners (ELLs) struggle with solving word problems for a number of reasons beyond math procedures or calculation challenges. As a result, ELLs may not only need math support but also reading and linguistic support. The purpose of this study was to assess the effectiveness of a math comprehension strategy called Dynamic Strategic…
ERIC Educational Resources Information Center
Seethaler, Pamela M.; Fuchs, Lynn S.; Fuchs, Douglas; Compton, Donald L.
2012-01-01
The purpose of this study was to assess the value of dynamic assessment (DA; degree of scaffolding required to learn unfamiliar mathematics content) for predicting 1st-grade calculations (CAs) and word problems (WPs) development, while controlling for the role of traditional assessments. Among 184 1st graders, predictors (DA, Quantity…
Predicting Development of Mathematical Word Problem Solving across the Intermediate Grades
ERIC Educational Resources Information Center
Tolar, Tammy D.; Fuchs, Lynn; Cirino, Paul T.; Fuchs, Douglas; Hamlett, Carol L.; Fletcher, Jack M.
2012-01-01
This study addressed predictors of the development of word problem solving (WPS) across the intermediate grades. At beginning of 3rd grade, 4 cohorts of students (N = 261) were measured on computation, language, nonverbal reasoning skills, and attentive behavior and were assessed 4 times from beginning of 3rd through end of 5th grade on 2 measures…
Teachers' Conceptions of Mathematical Word Problems: A Basis for Professional Development
ERIC Educational Resources Information Center
Chapman, Olive
2003-01-01
This paper reports on a study of mathematics teachers' thinking in the teaching of contextual or word problems [WP] with particular focus on teachers' conceptions of WP and the relationship to teaching. The 20 participants included Grades 1-12 preservice and inservice teachers. Data consisted of interviews and classroom observations. The findings…
Language and the Performance of English-Language Learners in Math Word Problems
ERIC Educational Resources Information Center
Martiniello, Maria
2008-01-01
In this article, Maria Martiniello reports the findings of a study of the linguistic complexity of math word problems that were found to exhibit differential item functioning for English-language learners (ELLs) and non-ELLs taking the Massachusetts Comprehensive Assessment System (MCAS) fourth-grade math test. It builds on prior research showing…
ERIC Educational Resources Information Center
Swanson, H. Lee; Moran, Amber S.; Bocian, Kathleen; Lussier, Cathy; Zheng, Xinhua
2013-01-01
This study investigated the role of generative strategies and working memory capacity on word problem solving accuracy in children with math difficulties (MD). Within classrooms, children in Grade 3 with MD ("n" = 69) were randomly assigned to one of three treatment conditions: paraphrase question propositions (Restate), paraphrase…
Word-Problem-Solving Strategy for Minority Students at Risk for Math Difficulties
ERIC Educational Resources Information Center
Kong, Jennifer E.; Orosco, Michael J.
2016-01-01
Minority students at risk for math difficulties (MD) struggle with word problems for various reasons beyond procedural or calculation challenges. As a result, these students require support in reading and language development in addition to math. The purpose of this study was to assess the effectiveness of a math comprehension strategy based on a…
Kindergartners' Understanding of Additive Commutativity within the Context of Word Problems.
ERIC Educational Resources Information Center
Wilkins, Jesse L. M.; Baroody, Arthur J.; Tiilikainen, Sirpa
2001-01-01
Investigated kindergartners' unary and binary understanding of additive commutativity using performance on tasks involving change-add-to and part-part-whole word problems, respectively. Found that data were inconsistent with models put forth by Baroody and Gannon and by Resnick and suggest three alternate theoretical explanations. Success on tasks…
ERIC Educational Resources Information Center
Lawrence, Virginia
No longer just a user of commercial software, the 21st century teacher is a designer of interactive software based on theories of learning. This software, a comprehensive study of straightline equations, enhances conceptual understanding, sketching, graphic interpretive and word problem solving skills as well as making connections to real-life and…
NASA Technical Reports Server (NTRS)
Ruge, J. W.; Stueben, K.
1987-01-01
The state of the art in algebraic multgrid (AMG) methods is discussed. The interaction between the relaxation process and the coarse grid correction necessary for proper behavior of the solution probes is discussed in detail. Sufficient conditions on relaxation and interpolation for the convergence of the V-cycle are given. The relaxation used in AMG, what smoothing means in an algebraic setting, and how it relates to the existing theory are considered. Some properties of the coarse grid operator are discussed, and results on the convergence of two-level and multilevel convergence are given. Details of an algorithm particularly studied for problems obtained by discretizing a single elliptic, second order partial differential equation are given. Results of experiments with such problems using both finite difference and finite element discretizations are presented.
Assessing Algebraic Solving Ability: A Theoretical Framework
ERIC Educational Resources Information Center
Lian, Lim Hooi; Yew, Wun Thiam
2012-01-01
Algebraic solving ability had been discussed by many educators and researchers. There exists no definite definition for algebraic solving ability as it can be viewed from different perspectives. In this paper, the nature of algebraic solving ability in terms of algebraic processes that demonstrate the ability in solving algebraic problem is…
ERIC Educational Resources Information Center
Dixon, Juli K.; Andreasen, Janet B.; Avila, Cheryl L.; Bawatneh, Zyad; Deichert, Deana L.; Howse, Tashana D.; Turner, Mercedes Sotillo
2014-01-01
A goal of this study was to examine elementary preservice teachers' (PSTs) ability to contextualize and decontextualize fraction subtraction by asking them to write word problems to represent fraction subtraction expressions and to choose prewritten word problems to support given fraction subtraction expressions. Three themes emerged from the…
Facilitating Case Reuse during Problem Solving in Algebra-Based Physics
ERIC Educational Resources Information Center
Mateycik, Frances Ann
2010-01-01
This research project investigates students' development of problem solving schemata while using strategies that facilitate the process of using solved examples to assist with a new problem (case reuse). Focus group learning interviews were used to explore students' perceptions and understanding of several problem solving strategies. Individual…
Fuchs, Lynn S.; Compton, Donald L.; Fuchs, Douglas; Powell, Sarah R.; Schumacher, Robin F.; Hamlett, Carol L.; Vernier, Emily; Namkung, Jessica M.; Vukovic, Rose K.
2012-01-01
The purpose of this study was to investigate the contributions of domain-general cognitive resources and different forms of arithmetic development to individual differences in pre-algebraic knowledge. Children (n=279; mean age=7.59 yrs) were assessed on 7 domain-general cognitive resources as well as arithmetic calculations and word problems at start of 2nd grade and on calculations, word problems, and pre-algebraic knowledge at end of 3rd grade. Multilevel path analysis, controlling for instructional effects associated with the sequence of classrooms in which students were nested across grades 2–3, indicated arithmetic calculations and word problems are foundational to pre-algebraic knowledge. Also, results revealed direct contributions of nonverbal reasoning and oral language to pre-algebraic knowledge, beyond indirect effects that are mediated via arithmetic calculations and word problems. By contrast, attentive behavior, phonological processing, and processing speed contributed to pre-algebraic knowledge only indirectly via arithmetic calculations and word problems. PMID:22409764
ERIC Educational Resources Information Center
Matthews, Paul G.; Atkinson, Richard C.
This paper reports an experiment designed to test theoretical relations among fast problem solving, more complex and slower problem solving, and research concerning fundamental memory processes. Using a cathode ray tube, subjects were presented with propositions of the form "Y is in list X" which they memorized. In later testing they were asked to…
ERIC Educational Resources Information Center
Ling, Gan We; Ghazali, Munirah
2007-01-01
This descriptive study was aimed at looking into how Primary 5 pupils solve pre-algebra problems concerning patterns and unknown quantities. Specifically, objectives of this study were to describe Primary 5 pupils' solution strategies, modes of representations and justifications in: (a) discovering, describing and using numerical and geometrical…
Word problems: a review of linguistic and numerical factors contributing to their difficulty
Daroczy, Gabriella; Wolska, Magdalena; Meurers, Walt Detmar; Nuerk, Hans-Christoph
2015-01-01
Word problems (WPs) belong to the most difficult and complex problem types that pupils encounter during their elementary-level mathematical development. In the classroom setting, they are often viewed as merely arithmetic tasks; however, recent research shows that a number of linguistic verbal components not directly related to arithmetic contribute greatly to their difficulty. In this review, we will distinguish three components of WP difficulty: (i) the linguistic complexity of the problem text itself, (ii) the numerical complexity of the arithmetic problem, and (iii) the relation between the linguistic and numerical complexity of a problem. We will discuss the impact of each of these factors on WP difficulty and motivate the need for a high degree of control in stimuli design for experiments that manipulate WP difficulty for a given age group. PMID:25883575
Facilitating case reuse during problem solving in algebra-based physics
NASA Astrophysics Data System (ADS)
Mateycik, Frances Ann
This research project investigates students' development of problem solving schemata while using strategies that facilitate the process of using solved examples to assist with a new problem (case reuse). Focus group learning interviews were used to explore students' perceptions and understanding of several problem solving strategies. Individual clinical interviews were conducted and quantitative examination data were collected to assess students' conceptual understanding, knowledge organization, and problem solving performance on a variety of problem tasks. The study began with a short one-time treatment of two independent, research-based strategies chosen to facilitate case reuse. Exploration of students' perceptions and use of the strategies lead investigators to select one of the two strategies to be implemented over a full semester of focus group interviews. The strategy chosen was structure mapping. Structure maps are defined as visual representations of quantities and their associations. They were created by experts to model the appropriate mental organization of knowledge elements for a given physical concept. Students were asked to use these maps as they were comfortable while problem solving. Data obtained from this phase of our study (Phase I) offered no evidence of improved problem solving schema. The 11 contact hour study was barely sufficient time for students to become comfortable using the maps. A set of simpler strategies were selected for their more explicit facilitation of analogical reasoning, and were used together during two more semester long focus group treatments (Phase II and Phase III of this study). These strategies included the use of a step-by-step process aimed at reducing cognitive load associated with mathematical procedure, direct reflection of principles involved in a given set of problems, and the direct comparison of problem pairs designed to be void of surface similarities (similar objects or object orientations) and sharing
Graphic and algebraic solutions of the discordant lead-uranium age problem
Stieff, L.R.; Stern, T.W.
1961-01-01
for the contaminating common Pb206 and Pb207. The linear relationships noted in this graphical procedure have been extended to plots of the mole ratios of total Pb206 U238 ( tN206 N238) vs. total Pb207 U235 ( tN207 N235). This modification permits the calculation of concordant ages for unaltered samples using only the Pb207 Pb206 ratio of the contaminating common lead. If isotopic data are available for two samples of the same age, x and y, from the same or related deposits or outcrops, graphs of the normalized difference ratios [ ( N206 N204)x - ( N206 N204)y ( N238 N204)x -( N238 N204)y] vs. [ ( N207 N204)x - ( N207 N204)y ( N235 N204)x -( N235 N204)y] can give concordant ages corrected for unknown amounts of a common lead with an unknown Pb207/ Pb206 ratio. (If thorium is absent the difference ratios may be normalized with the more abundant index isotope, Pb208.) Similar plots of tho normalized, difference ratios for three genetically related samples (x - y) and(x - z), will give concordant ages corrected, in addition, for either one unknown period of past alteration or initial contamination by an older generation of radiogenic lead of unknown Pb207/Pb206 ratio. Practical numerical solutions for many of tho concordant age calculations are not currently available. However, the algebraic equivalents of these new graphical methods give equations which may be programmed for computing machines. For geologically probable parameters the equations of higher order have two positive real roots that rapidly converge on the exact concordant ages corrected for original radiogenic lead and for loss or gain of lead or uranium. Modifications of these general age equations expanded only to the second degree have been derived for use with desk calculators. These graphical and algebraic methods clearly suggest both the type and minimum number of samples necessary for adequate mathematical analysis of discordant lead isotope age data. This mathematical treatment also makes it clear t
Using Dynamic Geometry and Computer Algebra Systems in Problem Based Courses for Future Engineers
ERIC Educational Resources Information Center
Tomiczková, Svetlana; Lávicka, Miroslav
2015-01-01
It is a modern trend today when formulating the curriculum of a geometric course at the technical universities to start from a real-life problem originated in technical praxis and subsequently to define which geometric theories and which skills are necessary for its solving. Nowadays, interactive and dynamic geometry software plays a more and more…
Teachers' Approaches towards Word Problem Solving: Elaborating or Restricting the Problem Context
ERIC Educational Resources Information Center
Depaepe, Fien; De Corte, Erik; Verschaffel, Lieven
2010-01-01
This contribution reports about a seven-month long video-based study in two regular Flemish sixth-grade mathematics classrooms. The focus is on teachers' approaches towards problem solving. In our analysis we distinguished between a paradigmatic-oriented (focus on the mathematical structure) and a narrative-oriented (focus on the contextual…
ERIC Educational Resources Information Center
Jitendra, Asha K.; Petersen-Brown, Shawna; Lein, Amy E.; Zaslofsky, Anne F.; Kunkel, Amy K.; Jung, Pyung-Gang; Egan, Andrea M.
2015-01-01
This study examined the quality of the research base related to strategy instruction priming the underlying mathematical problem structure for students with learning disabilities and those at risk for mathematics difficulties. We evaluated the quality of methodological rigor of 18 group research studies using the criteria proposed by Gersten et…
Danker, Jared F; Anderson, John R
2007-04-15
In naturalistic algebra problem solving, the cognitive processes of representation and retrieval are typically confounded, in that transformations of the equations typically require retrieval of mathematical facts. Previous work using cognitive modeling has associated activity in the prefrontal cortex with the retrieval demands of algebra problems and activity in the posterior parietal cortex with the transformational demands of algebra problems, but these regions tend to behave similarly in response to task manipulations (Anderson, J.R., Qin, Y., Sohn, M.-H., Stenger, V.A., Carter, C.S., 2003. An information-processing model of the BOLD response in symbol manipulation tasks. Psychon. Bull. Rev. 10, 241-261; Qin, Y., Carter, C.S., Silk, E.M., Stenger, A., Fissell, K., Goode, A., Anderson, J.R., 2004. The change of brain activation patterns as children learn algebra equation solving. Proc. Natl. Acad. Sci. 101, 5686-5691). With this study we attempt to isolate activity in these two regions by using a multi-step algebra task in which transformation (parietal) is manipulated in the first step and retrieval (prefrontal) is manipulated in the second step. Counter to our initial predictions, both brain regions were differentially active during both steps. We designed two cognitive models, one encompassing our initial assumptions and one in which both processes were engaged during both steps. The first model provided a poor fit to the behavioral and neural data, while the second model fit both well. This simultaneously emphasizes the strong relationship between retrieval and representation in mathematical reasoning and demonstrates that cognitive modeling can serve as a useful tool for understanding task manipulations in neuroimaging experiments. PMID:17355908
Danker, Jared F; Anderson, John R
2007-04-15
In naturalistic algebra problem solving, the cognitive processes of representation and retrieval are typically confounded, in that transformations of the equations typically require retrieval of mathematical facts. Previous work using cognitive modeling has associated activity in the prefrontal cortex with the retrieval demands of algebra problems and activity in the posterior parietal cortex with the transformational demands of algebra problems, but these regions tend to behave similarly in response to task manipulations (Anderson, J.R., Qin, Y., Sohn, M.-H., Stenger, V.A., Carter, C.S., 2003. An information-processing model of the BOLD response in symbol manipulation tasks. Psychon. Bull. Rev. 10, 241-261; Qin, Y., Carter, C.S., Silk, E.M., Stenger, A., Fissell, K., Goode, A., Anderson, J.R., 2004. The change of brain activation patterns as children learn algebra equation solving. Proc. Natl. Acad. Sci. 101, 5686-5691). With this study we attempt to isolate activity in these two regions by using a multi-step algebra task in which transformation (parietal) is manipulated in the first step and retrieval (prefrontal) is manipulated in the second step. Counter to our initial predictions, both brain regions were differentially active during both steps. We designed two cognitive models, one encompassing our initial assumptions and one in which both processes were engaged during both steps. The first model provided a poor fit to the behavioral and neural data, while the second model fit both well. This simultaneously emphasizes the strong relationship between retrieval and representation in mathematical reasoning and demonstrates that cognitive modeling can serve as a useful tool for understanding task manipulations in neuroimaging experiments.
NASA Astrophysics Data System (ADS)
Koldan, Jelena; Puzyrev, Vladimir; de la Puente, Josep; Houzeaux, Guillaume; Cela, José María
2014-06-01
We present an elaborate preconditioning scheme for Krylov subspace methods which has been developed to improve the performance and reduce the execution time of parallel node-based finite-element (FE) solvers for 3-D electromagnetic (EM) numerical modelling in exploration geophysics. This new preconditioner is based on algebraic multigrid (AMG) that uses different basic relaxation methods, such as Jacobi, symmetric successive over-relaxation (SSOR) and Gauss-Seidel, as smoothers and the wave front algorithm to create groups, which are used for a coarse-level generation. We have implemented and tested this new preconditioner within our parallel nodal FE solver for 3-D forward problems in EM induction geophysics. We have performed series of experiments for several models with different conductivity structures and characteristics to test the performance of our AMG preconditioning technique when combined with biconjugate gradient stabilized method. The results have shown that, the more challenging the problem is in terms of conductivity contrasts, ratio between the sizes of grid elements and/or frequency, the more benefit is obtained by using this preconditioner. Compared to other preconditioning schemes, such as diagonal, SSOR and truncated approximate inverse, the AMG preconditioner greatly improves the convergence of the iterative solver for all tested models. Also, when it comes to cases in which other preconditioners succeed to converge to a desired precision, AMG is able to considerably reduce the total execution time of the forward-problem code-up to an order of magnitude. Furthermore, the tests have confirmed that our AMG scheme ensures grid-independent rate of convergence, as well as improvement in convergence regardless of how big local mesh refinements are. In addition, AMG is designed to be a black-box preconditioner, which makes it easy to use and combine with different iterative methods. Finally, it has proved to be very practical and efficient in the
Jitendra, Asha K; Petersen-Brown, Shawna; Lein, Amy E; Zaslofsky, Anne F; Kunkel, Amy K; Jung, Pyung-Gang; Egan, Andrea M
2015-01-01
This study examined the quality of the research base related to strategy instruction priming the underlying mathematical problem structure for students with learning disabilities and those at risk for mathematics difficulties. We evaluated the quality of methodological rigor of 18 group research studies using the criteria proposed by Gersten et al. and 10 single case design (SCD) research studies using criteria suggested by Horner et al. and the What Works Clearinghouse. Results indicated that 14 group design studies met the criteria for high-quality or acceptable research, whereas SCD studies did not meet the standards for an evidence-based practice. Based on these findings, strategy instruction priming the mathematics problem structure is considered an evidence-based practice using only group design methodological criteria. Implications for future research and for practice are discussed.
Thevenot, Catherine; Devidal, Michel; Barrouillet, Pierre; Fayol, Michel
2007-01-01
The aim of this paper is to investigate the controversial issue of the nature of the representation constructed by individuals to solve arithmetic word problems. More precisely, we consider the relevance of two different theories: the situation or mental model theory (Johnson-Laird, 1983; Reusser, 1989) and the schema theory (Kintsch & Greeno, 1985; Riley, Greeno, & Heller, 1983). Fourth-graders who differed in their mathematical skills were presented with problems that varied in difficulty and with the question either before or after the text. We obtained the classic effect of the position of the question, with better performance when the question was presented prior to the text. In addition, this effect was more marked in the case of children who had poorer mathematical skills and in the case of more difficult problems. We argue that this pattern of results is compatible only with the situation or mental model theory, and not with the schema theory. PMID:17162507
ERIC Educational Resources Information Center
Schmidt, Sylvine; Bednarz, Nadine
1997-01-01
Discusses the difficulties observed in the transition from teaching arithmetic to teaching algebra. Future teachers (n=164) were questioned regarding to what extent they were able to shift back and forth between teaching methods within the context of problem solving. Interviews were conducted individually and in a dyad format. (AIM)
SO(4) algebraic approach to the three-body bound state problem in two dimensions
NASA Astrophysics Data System (ADS)
Dmitrašinović, V.; Salom, Igor
2014-08-01
We use the permutation symmetric hyperspherical three-body variables to cast the non-relativistic three-body Schrödinger equation in two dimensions into a set of (possibly decoupled) differential equations that define an eigenvalue problem for the hyper-radial wave function depending on an SO(4) hyper-angular matrix element. We express this hyper-angular matrix element in terms of SO(3) group Clebsch-Gordan coefficients and use the latter's properties to derive selection rules for potentials with different dynamical/permutation symmetries. Three-body potentials acting on three identical particles may have different dynamical symmetries, in order of increasing symmetry, as follows: (1) S3 ⊗ OL(2), the permutation times rotational symmetry, that holds in sums of pairwise potentials, (2) O(2) ⊗ OL(2), the so-called "kinematic rotations" or "democracy symmetry" times rotational symmetry, that holds in area-dependent potentials, and (3) O(4) dynamical hyper-angular symmetry, that holds in hyper-radial three-body potentials. We show how the different residual dynamical symmetries of the non-relativistic three-body Hamiltonian lead to different degeneracies of certain states within O(4) multiplets.
Peake, Christian; Jiménez, Juan E; Rodríguez, Cristina; Bisschop, Elaine; Villarroel, Rebeca
2015-01-01
Arithmetic word problem (AWP) solving is a highly demanding task for children with learning disabilities (LD) since verbal and mathematical information have to be integrated. This study examines specifically how syntactic awareness (SA), the ability to manage the grammatical structures of language, affects AWP solving. Three groups of children in elementary education were formed: children with arithmetic learning disabilities (ALD), children with reading learning disabilities (RLD), and children with comorbid arithmetic and reading learning disabilities (ARLD). Mediation analysis confirmed that SA was a mediator variable for both groups of children with reading disabilities when solving AWPs, but not for children in the ALD group. All groups performed below the control group in the problem solving task. When SA was controlled for, semantic structure and position of the unknown set were variables that affected both groups with ALD. Specifically, children with ALD only were more affected by the place of the unknown set. PMID:24509567
Peake, Christian; Jiménez, Juan E; Rodríguez, Cristina; Bisschop, Elaine; Villarroel, Rebeca
2015-01-01
Arithmetic word problem (AWP) solving is a highly demanding task for children with learning disabilities (LD) since verbal and mathematical information have to be integrated. This study examines specifically how syntactic awareness (SA), the ability to manage the grammatical structures of language, affects AWP solving. Three groups of children in elementary education were formed: children with arithmetic learning disabilities (ALD), children with reading learning disabilities (RLD), and children with comorbid arithmetic and reading learning disabilities (ARLD). Mediation analysis confirmed that SA was a mediator variable for both groups of children with reading disabilities when solving AWPs, but not for children in the ALD group. All groups performed below the control group in the problem solving task. When SA was controlled for, semantic structure and position of the unknown set were variables that affected both groups with ALD. Specifically, children with ALD only were more affected by the place of the unknown set.
ERIC Educational Resources Information Center
Pavelle, Richard; And Others
1981-01-01
Describes the nature and use of computer algebra and its applications to various physical sciences. Includes diagrams illustrating, among others, a computer algebra system and flow chart of operation of the Euclidean algorithm. (SK)
NASA Astrophysics Data System (ADS)
Banerjee, Banmali
Methods and procedures for successfully solving math word problems have been, and continue to be a mystery to many U.S. high school students. Previous studies suggest that the contextual and mathematical understanding of a word problem, along with the development of schemas and their related external representations, positively contribute to students' accomplishments when solving word problems. Some studies have examined the effects of diagramming on students' abilities to solve word problems that only involved basic arithmetic operations. Other studies have investigated how instructional models that used technology influenced students' problem solving achievements. Still other studies have used schema-based instruction involving students with learning disabilities. No study has evaluated regular high school students' achievements in solving standard math word problems using a diagramming technique without technological aid. This study evaluated students' achievement in solving math word problems using a diagramming technique. Using a quasi-experimental experimental pretest-posttest research design, quantitative data were collected from 172 grade 11 Hispanic English language learners (ELLS) and African American learners whose first language is English (EFLLs) in 18 classes at an inner city high school in Northern New Jersey. There were 88 control and 84 experimental students. The pretest and posttest of each participating student and samples of the experimental students' class assignments provided the qualitative data for the study. The data from this study exhibited that the diagramming method of solving math word problems significantly improved student achievement in the experimental group (p<.01) compared to the control group. The study demonstrated that urban, high school, ELLs benefited from instruction that placed emphasis on the mathematical vocabulary and symbols used in word problems and that both ELLs and EFLLs improved their problem solving success
ERIC Educational Resources Information Center
Reusser, Kurt; And Others
The main concern of this paper is on the psychological processes of how students understand and solve mathematical word problems, and on how this knowledge can be applied to computer-based tutoring. It is argued that only a better understanding of the psychological requirements for understanding and solving those problems will lead to…
NASA Astrophysics Data System (ADS)
Connes, Alain; Kreimer, Dirk
This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra which is commutative as an algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of . We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop
ERIC Educational Resources Information Center
Abdullah, Nasarudin; Halim, Lilia; Zakaria, Effandi
2014-01-01
This study aimed to determine the impact of strategic thinking and visual representation approaches (VStops) on the achievement, conceptual knowledge, metacognitive awareness, awareness of problem-solving strategies, and student attitudes toward mathematical word problem solving among primary school students. The experimental group (N = 96)…
ERIC Educational Resources Information Center
Kribbs, Elizabeth E.; Rogowsky, Beth A.
2016-01-01
Mathematics word-problems continue to be an insurmountable challenge for many middle school students. Educators have used pictorial and schematic illustrations within the classroom to help students visualize these problems. However, the data shows that pictorial representations can be more harmful than helpful in that they only display objects or…
ERIC Educational Resources Information Center
Dennis, Minyi Shih; Knight, Jacqueline; Jerman, Olga
2016-01-01
This article describes how to teach fraction and percentage word problems using a model-drawing strategy. This cognitive strategy places emphasis on explicitly teaching students how to draw a schematic diagram to represent the qualitative relations described in the problem, and how to formulate the solution based on the schematic diagram. The…
ERIC Educational Resources Information Center
de Kock, Willem D.; Harskamp, Egbert G.
2014-01-01
Teachers in primary education experience difficulties in teaching word problem solving in their mathematics classes. However, during controlled experiments with a metacognitive computer programme, students' problem-solving skills improved. Also without the supervision of researchers, metacognitive computer programmes can be beneficial in a…
ERIC Educational Resources Information Center
Swanson, H. Lee; Lussier, Catherine; Orosco, Michael
2011-01-01
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…
ERIC Educational Resources Information Center
Patkin, Dorit; Gazit, Avikam
2011-01-01
This article aims to present the findings of a research which investigated the effect of a difference in word formulation and mathematical characteristics of story problems on their successful solution by preservice mathematics teachers (students) and practising mathematics teachers. The findings show that in the case of a problem with a…
ERIC Educational Resources Information Center
Manalo, Emmanuel; Uesaka, Yuri
2006-01-01
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…
García, Ana I; Jiménez, Juan E; Hess, Stephany
2006-01-01
This study was designed to determine a word problem difficulty classification in children with arithmetic learning disabilities (ALD; n = 104) in comparison with typically achieving students (n = 44). We tested variables such as (a) semantic structure (Change, Combine, Compare, and Equalize), (b) operation (subtraction and addition), and (c) position of the unknown quantity in the problem. Facet theory with multidimensional scaling techniques (MINISSA) was used to analyze the underlying dimensions in the responses of each group of participants. Our results indicate that although the word problem difficulty classifications for the 2 groups of children were different, the position of the unknown quantity had a greater influence on the level of difficulty of story problems than other variables. The noncanonical problems--specifically, those with the unknown term in the first place--although difficult for both groups of children, were the most difficult problems for children with ALD.
Predicting Development of Mathematical Word Problem Solving Across the Intermediate Grades.
Tolar, Tammy D; Fuchs, Lynn; Cirino, Paul T; Fuchs, Douglas; Hamlett, Carol L; Fletcher, Jack M
2012-01-01
This study addressed predictors of the development of word problem solving (WPS) across the intermediate grades. At beginning of 3rd grade, 4 cohorts of students (N = 261) were measured on computation, language, nonverbal reasoning skills, and attentive behavior and were assessed 4 times from beginning of 3rd through end of 5th grade on 2 measures of WPS at low and high levels of complexity. Language skills were related to initial performance at both levels of complexity and did not predict growth at either level. Computational skills had an effect on initial performance in low- but not high-complexity problems and did not predict growth at either level of complexity. Attentive behavior did not predict initial performance but did predict growth in low-complexity, whereas it predicted initial performance but not growth for high-complexity problems. Nonverbal reasoning predicted initial performance and growth for low-complexity WPS, but only growth for high-complexity WPS. This evidence suggests that although mathematical structure is fixed, different cognitive resources may act as limiting factors in WPS development when the WPS context is varied.
ERIC Educational Resources Information Center
Erktin, Emine; Akyel, Ayse
2005-01-01
Mathematics educators are concerned about students' lack of ability to translate mathematical word problems into computable forms. Researchers argue that linguistic problems lie at the root of students' difficulties with mathematical word problems. The issue becomes more complicated for bilingual students. It is argued that if students study…
NASA Astrophysics Data System (ADS)
Nara, T.; Koiwa, K.; Takagi, S.; Oyama, D.; Uehara, G.
2014-05-01
This paper presents an algebraic reconstruction method for dipole-quadrupole sources using magnetoencephalography data. Compared to the conventional methods with the equivalent current dipoles source model, our method can more accurately reconstruct two close, oppositely directed sources. Numerical simulations show that two sources on both sides of the longitudinal fissure of cerebrum are stably estimated. The method is verified using a quadrupolar source phantom, which is composed of two isosceles-triangle-coils with parallel bases.
Using PROC GLIMMIX to Analyze the Animal Watch, a Web-Based Tutoring System for Algebra Readiness
ERIC Educational Resources Information Center
Barbu, Otilia C.
2012-01-01
In this study, I investigated how proficiently seventh-grade students enrolled in two Southwestern schools solve algebra word problems. I analyzed various factors that could affect this proficiency and explored the differences between English Learners (ELs) and native English Primary students (EPs). I collected the data as part of the Animal Watch…
ERIC Educational Resources Information Center
Gallardo, Aurora
2002-01-01
Analyzes from an historical perspective the extension of the natural-number domain to integers in students' transition from arithmetic to algebra in the context of word problems. Extracts four levels of acceptance of these numbers--subtrahend, relative number, isolated number and formal negative number--from historical texts. The first three…
ERIC Educational Resources Information Center
Swanson, H. Lee
2014-01-01
Cognitive strategies are important tools for children with math difficulties (MD) in learning to solve word problems. The effectiveness of strategy training, however, depends on working memory capacity (WMC). Thus, children with MD but with relatively higher WMC are more likely to benefit from strategy training, whereas children with lower WMC may…
ERIC Educational Resources Information Center
Powell, Sarah R.; Fuchs, Lynn S.; Cirino, Paul T.; Fuchs, Douglas; Compton, Donald L.; Changas, Paul C.
2015-01-01
The focus of the present study was enhancing word problem and calculation achievement in ways that support prealgebraic thinking among second-grade students at risk for mathematics difficulty. Intervention relied on a multitier support system (i.e., responsiveness to intervention, or RTI) in which at-risk students participate in general classroom…
ERIC Educational Resources Information Center
Fuchs, Lynn S.; Schumacher, Robin F.; Long, Jessica; Namkung, Jessica; Malone, Amelia S.; Wang, Amber; Hamlett, Carol L.; Jordan, Nancy C.; Siegler, Robert S.; Changas, Paul
2016-01-01
The purposes of this study were to (a) investigate the efficacy of a core fraction intervention program on understanding and calculation skill and (b) isolate the effects of different forms of fraction word-problem (WP) intervention. At-risk fourth graders (n = 213) were randomly assigned to the school's business-as-usual program, or one of two…
ERIC Educational Resources Information Center
Fuchs, Lynn S.; Schumacher, Robin F.; Long, Jessica; Namkung, Jessica; Malone, Amelia S.; Wang, Amber; Hamlett, Carol L.; Jordan, Nancy C.; Siegler, Robert S.; Changas, Paul
2016-01-01
The purposes of this study were to (a) investigate the efficacy of a core fraction intervention program on understanding and calculation skill and (b) isolate the effects of different forms of fraction word-problem (WP) intervention delivered as part of the larger program. At-risk 4th graders (n = 213) were randomly assigned at the individual…
ERIC Educational Resources Information Center
Awofala, Adeneye O. A.
2014-01-01
This study investigated the effect of a personalised print-based instruction versus a non-personalised print-based instruction on the attitudes toward mathematics word problems of 350 senior secondary school year one Nigerian students within the blueprint of a quantitative research of pre-treatment-intervention-post-treatment non-equivalent…
ERIC Educational Resources Information Center
Berberoglu, Giray
1995-01-01
Item characteristic curves were compared across gender and socioeconomic status (SES) groups for the university entrance mathematics examination in Turkey to see if any group had an advantage in solving computation, word-problem, or geometry questions. Differential item functioning was found, and patterns are discussed. (SLD)
ERIC Educational Resources Information Center
Garcia, Ana I.; Jimenez, Juan E.; Hess, Stephany
2006-01-01
This study was designed to determine a word problem difficulty classification in children with arithmetic learning disabilities (ALD; n = 104) in comparison with typically achieving students (n = 44). We tested variables such as (a) semantic structure (Change, Combine, Compare, and Equalize), (b) operation (subtraction and addition), and (c)…
ERIC Educational Resources Information Center
Reikeras, Elin K. L.
2009-01-01
Performance in consistent arithmetical word problems was assessed in 941 pupils aged eight (N = 415), ten (N = 274), and thirteen (N = 252) classified in four achievement groups by standardised achievement tests: low achievement in both mathematics and reading (MLRL), in mathematics only (ML-only), in reading only (RL-only), and normal achievement…
ERIC Educational Resources Information Center
Chan, Simon
2015-01-01
In learning mathematics through English, one of the major challenges facing English as a Foreign Language (EFL) learners is understanding the language used to present word problems in mathematics texts. Without comprehending such language, learners are not able to carry out the targeted calculations no matter how familiar they are with the…
ERIC Educational Resources Information Center
Dewolf, Tinne; Van Dooren, Wim; Hermens, Frouke; Verschaffel, Lieven
2015-01-01
During the last two decades various researchers confronted upper elementary and lower secondary school pupils with word problems that were problematic from a realistic modelling point of view (so-called P-items), and found that pupils in general did not use their everyday knowledge to solve such P-items. Several attempts were undertaken to…
ERIC Educational Resources Information Center
Schaufele, Christopher; Zumoff, Nancy
Earth Algebra is an entry level college algebra course that incorporates the spirit of the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics at the college level. The context of the course places mathematics at the center of one of the major current concerns of the world. Through…
ERIC Educational Resources Information Center
Cavanagh, Sean
2009-01-01
As educators and policymakers search for ways to prepare students for the rigors of algebra, teachers in the Helena, Montana, school system are starting early by attempting to nurture students' algebraic-reasoning ability, as well as their basic number skills, in early elementary school, rather than waiting until middle or early high school.…
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumps and an anisotropy in one part.
One Language, Two Number-Word Systems and Many Problems: Numerical Cognition in the Czech Language
ERIC Educational Resources Information Center
Pixner, S.; Zuber, J.; Hermanova, V.; Kaufmann, L.; Nuerk, H.-C.; Moeller, K.
2011-01-01
Comparing numerical performance between different languages does not only mean comparing different number-word systems, but also implies a comparison of differences regarding culture or educational systems. The Czech language provides the remarkable opportunity to disentangle this confound as there exist two different number-word systems within…
ERIC Educational Resources Information Center
Jitendra, Asha; Xin, Yan Ping
1997-01-01
Summarizes 14 mathematics word-problem-solving intervention studies for elementary and secondary students with disabilities or who were at-risk for math failure. Interventions included representational techniques (diagramming), strategy-training procedures (cognitive and metacognitive), task variations (sequencing and word-problem context), and…
ERIC Educational Resources Information Center
Jitendra, Asha K.; Corroy, Kelly Cozine; Dupuis, Danielle N.
2013-01-01
The purposes of this study were (a) to evaluate differences in arithmetic word problem solving between high and low at-risk students for mathematics difficulties (MD) and (b) to assess the influence of attention, behavior, reading, and socio-economic status (SES) in predicting the word problem solving performance of third-grade students with MD.…
ERIC Educational Resources Information Center
Nortvedt, Guri A.
2011-01-01
This article discusses how 13-year-old students with above-average numeracy skills and below-average reading skills cope with comprehending word problems. Compared to other students who are proficient in numeracy and are skilled readers, these students are more disadvantaged when solving single-step and multistep arithmetic word problems. The…
ERIC Educational Resources Information Center
Csikos, Csaba; Szitanyi, Judit; Kelemen, Rita
2012-01-01
The present study aims to investigate the effects of a design experiment developed for third-grade students in the field of mathematics word problems. The main focus of the program was developing students' knowledge about word problem solving strategies with an emphasis on the role of visual representations in mathematical modeling. The experiment…
Patterns to Develop Algebraic Reasoning
ERIC Educational Resources Information Center
Stump, Sheryl L.
2011-01-01
What is the role of patterns in developing algebraic reasoning? This important question deserves thoughtful attention. In response, this article examines some differing views of algebraic reasoning, discusses a controversy regarding patterns, and describes how three types of patterns--in contextual problems, in growing geometric figures, and in…
The Effect of Strategy on Problem Solving: An FMRI Study
ERIC Educational Resources Information Center
Newman, Sharlene D.; Pruce, Benjamin; Rusia, Akash; Burns, Thomas, Jr.
2010-01-01
fMRI was used to examine the differential effect of two problem-solving strategies. Participants were trained to use both a pictorial/spatial and a symbolic/algebraic strategy to solve word problems. While these two strategies activated similar cortical regions, a number of differences were noted in the level of activation. These differences…
Algebra from Chips and Chopsticks
ERIC Educational Resources Information Center
Yun, Jeong Oak; Flores, Alfinio
2012-01-01
Students can use geometric representations of numbers as a way to explore algebraic ideas. With the help of these representations, students can think about the relations among the numbers, express them using their own words, and represent them with letters. The activities discussed here can stimulate students to try to find various ways of solving…
Twining characters and orbit Lie algebras
Fuchs, Jurgen; Ray, Urmie; Schellekens, Bert; Schweigert, Christoph
1996-12-05
We associate to outer automorphisms of generalized Kac-Moody algebras generalized character-valued indices, the twining characters. A character formula for twining characters is derived which shows that they coincide with the ordinary characters of some other generalized Kac-Moody algebra, the so-called orbit Lie algebra. Some applications to problems in conformal field theory, algebraic geometry and the theory of sporadic simple groups are sketched.
Computer Algebra versus Manipulation
ERIC Educational Resources Information Center
Zand, Hossein; Crowe, David
2004-01-01
In the UK there is increasing concern about the lack of skill in algebraic manipulation that is evident in students entering mathematics courses at university level. In this note we discuss how the computer can be used to ameliorate some of the problems. We take as an example the calculations needed in three dimensional vector analysis in polar…
ERIC Educational Resources Information Center
Glick, David
1995-01-01
Presents a technique that helps students concentrate more on the science and less on the mechanics of algebra while dealing with introductory physics formulas. Allows the teacher to do complex problems at a lower level and not be too concerned about the mathematical abilities of the students. (JRH)
ERIC Educational Resources Information Center
Ormond, Christine
2012-01-01
Primary teachers play a key role in their students' future mathematical success in the early secondary years. While the word "algebra" may make some primary teachers feel uncomfortable or worried, the basic arithmetic ideas underlying algebra are vitally important for older primary students as they are increasingly required to use "algebraic…
Pseudo Algebraically Closed Extensions
NASA Astrophysics Data System (ADS)
Bary-Soroker, Lior
2009-07-01
This PhD deals with the notion of pseudo algebraically closed (PAC) extensions of fields. It develops a group-theoretic machinery, based on a generalization of embedding problems, to study these extensions. Perhaps the main result is that although there are many PAC extensions, the Galois closure of a proper PAC extension is separably closed. The dissertation also contains the following subjects. The group theoretical counterpart of pseudo algebraically closed extensions, the so-called projective pairs. Applications to seemingly unrelated subjects, e.g., an analog of Dirichlet's theorem about primes in arithmetic progression for polynomial rings in one variable over infinite fields.
Some C∗-algebras which are coronas of non-C∗-Banach algebras
NASA Astrophysics Data System (ADS)
Voiculescu, Dan-Virgil
2016-07-01
We present results and motivating problems in the study of commutants of hermitian n-tuples of Hilbert space operators modulo normed ideals. In particular, the C∗-algebras which arise in this context as coronas of non-C∗-Banach algebras, the connections with normed ideal perturbations of operators, the hyponormal operators and the bidual Banach algebras one encounters are discussed.
Generalization of n-ary Nambu algebras and beyond
Ataguema, H.; Makhlouf, A.; Silvestrov, S.
2009-08-15
The aim of this paper is to introduce n-ary Hom-algebra structures generalizing the n-ary algebras of Lie type including n-ary Nambu algebras, n-ary Nambu-Lie algebras and n-ary Lie algebras, and n-ary algebras of associative type including n-ary totally associative and n-ary partially associative algebras. We provide examples of the new structures and present some properties and construction theorems. We describe the general method allowing one to obtain an n-ary Hom-algebra structure starting from an n-ary algebra and an n-ary algebra endomorphism. Several examples are derived using this process. Also we initiate investigation of classification problems for algebraic structures introduced in the article and describe all ternary three-dimensional Hom-Nambu-Lie structures with diagonal homomorphism.
ERIC Educational Resources Information Center
Schauble, Leona; Peel, Tina
Problem solving is a main topic in mathematics education, and considerable headway has been made in identifying the processes involved in solving well-formed problems like algebra word problems, mathematical algorithms, and logical puzzles like the Tower of Hanoi. The "Mathnet" format of the SQUARE ONE TV program, however, requires viewers to…
2013-05-06
AMG2013 is a parallel algebraic multigrid solver for linear systems arising from problems on unstructured grids. It has been derived directly from the Boomer AMG solver in the hypre library, a large linear solvers library that is being developed in the Center for Applied Scientific Computing (CASC) at LLNL. The driver provided in the benchmark can build various test problems. The default problem is a Laplace type problem on an unstructured domain with various jumpsmore » and an anisotropy in one part.« less
Inequalities, Assessment and Computer Algebra
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2015-01-01
The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in…
Words, Words, Words: English, Vocabulary.
ERIC Educational Resources Information Center
Lamb, Barbara
The Quinmester course on words gives the student the opportunity to increase his proficiency by investigating word origins, word histories, morphology, and phonology. The course includes the following: dictionary skills and familiarity with the "Oxford,""Webster's Third," and "American Heritage" dictionaries; word derivations from other languages;…
Lee, Jaehoon; Wilczek, Frank
2013-11-27
Motivated by the problem of identifying Majorana mode operators at junctions, we analyze a basic algebraic structure leading to a doubled spectrum. For general (nonlinear) interactions the emergent mode creation operator is highly nonlinear in the original effective mode operators, and therefore also in the underlying electron creation and destruction operators. This phenomenon could open up new possibilities for controlled dynamical manipulation of the modes. We briefly compare and contrast related issues in the Pfaffian quantum Hall state.
Literal algebra for satellite dynamics. [perturbation analysis
NASA Technical Reports Server (NTRS)
Gaposchkin, E. M.
1975-01-01
A description of the rather general class of operations available is given and the operations are related to problems in satellite dynamics. The implementation of an algebra processor is discussed. The four main categories of symbol processors are related to list processing, string manipulation, symbol manipulation, and formula manipulation. Fundamental required operations for an algebra processor are considered. It is pointed out that algebra programs have been used for a number of problems in celestial mechanics with great success. The advantage of computer algebra is its accuracy and speed.
ERIC Educational Resources Information Center
Despina, Desli; Harikleia, Loukidou
2014-01-01
Mathematics textbooks are a predominant resource in primary school in Greece, as well as in many other countries. The present study reports on both a content analysis of Greek mathematics textbooks with regard to the types of word problems represented in them and a quantitative analysis of children's achievement in these problems. For the…
ERIC Educational Resources Information Center
Leh, Jayne M.; Jitendra, Asha K.
2013-01-01
This study compared the effectiveness of computer-mediated instruction (CMI) and teacher-mediated instruction (TMI) on the word problem-solving performance of students struggling in mathematics. Both conditions integrated cognitive modeling that focused on the problem structure using visual representations with critical instructional elements…
Oostermeijer, Meike; Boonen, Anton J. H.; Jolles, Jelle
2014-01-01
The scientific literature shows that constructive play activities are positively related to children’s spatial ability. Likewise, a close positive relation is found between spatial ability and mathematical word problem-solving performances. The relation between children’s constructive play and their performance on mathematical word problems is, however, not reported yet. The aim of the present study was to investigate whether spatial ability acted as a mediator in the relation between constructive play and mathematical word problem-solving performance in 128 sixth-grade elementary school children. This mediating role of spatial ability was tested by utilizing the current mediation approaches suggested by Preacher and Hayes (2008). Results showed that 38.16% of the variance in mathematical word problem-solving performance is explained by children’s constructive play activities and spatial ability. More specifically, spatial ability acted as a partial mediator, explaining 31.58% of the relation between constructive play and mathematical word problem-solving performance. PMID:25101038
Sheriff, Kelli A; Boon, Richard T
2014-08-01
The purpose of this study was to examine the effects of computer-based graphic organizers, using Kidspiration 3© software, to solve one-step word problems. Participants included three students with mild intellectual disability enrolled in a functional academic skills curriculum in a self-contained classroom. A multiple probe single-subject research design (Horner & Baer, 1978) was used to evaluate the effectiveness of computer-based graphic organizers to solving mathematical one-step word problems. During the baseline phase, the students completed a teacher-generated worksheet that consisted of nine functional word problems in a traditional format using a pencil, paper, and a calculator. In the intervention and maintenance phases, the students were instructed to complete the word problems using a computer-based graphic organizer. Results indicated that all three of the students improved in their ability to solve the one-step word problems using computer-based graphic organizers compared to traditional instructional practices. Limitations of the study and recommendations for future research directions are discussed.
NASA Astrophysics Data System (ADS)
Vaninsky, Alexander
2011-04-01
This article introduces a trigonometric field (TF) that extends the field of real numbers by adding two new elements: sin and cos - satisfying an axiom sin2 + cos2 = 1. It is shown that by assigning meaningful names to particular elements of the field, all known trigonometric identities may be introduced and proved. Two different interpretations of the TF are discussed with many others potentially possible. The main objective of this article is to introduce a broader view of trigonometry that can serve as motivation for mathematics students and teachers to study and teach abstract algebraic structures.
ERIC Educational Resources Information Center
Hills, George L. C.
1981-01-01
Based on information gained in an interview with a 12-year-old girl in grade seven, a rational reconstruction of the student's problem-solving strategy is proposed and compared with the strategies normally prescribed in contemporary school mathematics textbooks. (Author/CHC)
ERIC Educational Resources Information Center
Goodwin, Amanda P.
2016-01-01
This study explores the effectiveness of integrating morphological instruction within comprehension strategy instruction. Participants were 203 students (N = 117 fifth-grade; 86 sixth-grade) from four urban schools who were randomly assigned to the intervention (N = 110; morphological problem-solving within comprehension strategy instruction) or…
MONTO: A Machine-Readable Ontology for Teaching Word Problems in Mathematics
ERIC Educational Resources Information Center
Lalingkar, Aparna; Ramnathan, Chandrashekar; Ramani, Srinivasan
2015-01-01
The Indian National Curriculum Framework has as one of its objectives the development of mathematical thinking and problem solving ability. However, recent studies conducted in Indian metros have expressed concern about students' mathematics learning. Except in some private coaching academies, regular classroom teaching does not include problem…
ERIC Educational Resources Information Center
Schoppek, Wolfgang; Tulis, Maria
2010-01-01
The fluency of basic arithmetical operations is a precondition for mathematical problem solving. However, the training of skills plays a minor role in contemporary mathematics instruction. The authors proposed individualization of practice as a means to improve its efficiency, so that the time spent with the training of skills is minimized. As a…
ERIC Educational Resources Information Center
Pacheco, Mark B.; Goodwin, Amanda P.
2013-01-01
Adolescents often use root word and affix knowledge to figure out unknown words. Anglin (1993) found that younger readers favor the Part-to-Whole strategy, and Tyler and Nagy (1989) confirmed the importance of root-word knowledge for middle school students. This study seeks to understand the different strategies middle school readers use so that…
Using the Internet To Investigate Algebra.
ERIC Educational Resources Information Center
Sherwood, Walter
The lesson plans in this book engage students by using a tool they enjoy--the Internet--to explore key concepts in algebra. Working either individually or in groups, students learn to approach algebra from a problem solving perspective. Each lesson shows learners how to use the Internet as a resource for gathering facts, data, and other…
Algebraic Geodesics on Three-Dimensional Quadrics
NASA Astrophysics Data System (ADS)
Kai, Yue
2015-12-01
By Hamilton-Jacobi method, we study the problem of algebraic geodesics on the third-order surface. By the implicit function theorem, we proved the existences of the real geodesics which are the intersections of two algebraic surfaces, and we also give some numerical examples.
Swanson, H. Lee
2015-01-01
This study investigated the role of strategy instruction and working memory capacity (WMC) on problem solving solution accuracy in children with and without math disabilities (MD). Children in grade 3 (N = 204) with and without MD subdivided into high and low WMC were randomly assigned to 1 of 4 conditions: verbal strategies (e.g., underlining question sentence), visual strategies (e.g., correctly placing numbers in diagrams), verbal + visual strategies, and an untreated control. The dependent measures for training were problem solving accuracy and two working memory transfer measures (operation span and visual-spatial span). Three major findings emerged: (1) strategy instruction facilitated solution accuracy but the effects of strategy instruction were moderated by WMC, (2) some strategies yielded higher post-test scores than others, but these findings were qualified as to whether children were at risk for MD, and (3) strategy training on problem solving measures facilitated transfer to working memory measures. The main findings were that children with MD, but high WM spans, were more likely to benefit from strategy conditions on target and transfer measures than children with lower WMC. The results suggest that WMC moderates the influence of cognitive strategies on both the targeted and non-targeted measures. PMID:26300803
Lai, Yinghui; Zhu, Xiaoshuang; Chen, Yinghe; Li, Yanjun
2015-01-01
Mathematics is one of the most objective, logical, and practical academic disciplines. Yet, in addition to cognitive skills, mathematical problem solving also involves affective factors. In the current study, we first investigated effects of mathematics anxiety (MA) and mathematical metacognition on word problem solving (WPS). We tested 224 children (116 boys, M = 10.15 years old, SD = 0.56) with the Mathematics Anxiety Scale for Children, the Chinese Revised-edition Questionnaire of Pupil's Metacognitive Ability in Mathematics, and WPS tasks. The results indicated that mathematical metacognition mediated the effect of MA on WPS after controlling for IQ. Second, we divided the children into four mathematics achievement groups including high achieving (HA), typical achieving (TA), low achieving (LA), and mathematical learning difficulty (MLD). Because mathematical metacognition and MA predicted mathematics achievement, we compared group differences in metacognition and MA with IQ partialled out. The results showed that children with MLD scored lower in self-image and higher in learning mathematics anxiety (LMA) than the TA and HA children, but not in mathematical evaluation anxiety (MEA). MLD children's LMA was also higher than that of their LA counterparts. These results provide insight into factors that may mediate poor WPS performance which emerges under pressure in mathematics. These results also suggest that the anxiety during learning mathematics should be taken into account in mathematical learning difficulty interventions.
Lai, Yinghui; Zhu, Xiaoshuang; Chen, Yinghe; Li, Yanjun
2015-01-01
Mathematics is one of the most objective, logical, and practical academic disciplines. Yet, in addition to cognitive skills, mathematical problem solving also involves affective factors. In the current study, we first investigated effects of mathematics anxiety (MA) and mathematical metacognition on word problem solving (WPS). We tested 224 children (116 boys, M = 10.15 years old, SD = 0.56) with the Mathematics Anxiety Scale for Children, the Chinese Revised-edition Questionnaire of Pupil’s Metacognitive Ability in Mathematics, and WPS tasks. The results indicated that mathematical metacognition mediated the effect of MA on WPS after controlling for IQ. Second, we divided the children into four mathematics achievement groups including high achieving (HA), typical achieving (TA), low achieving (LA), and mathematical learning difficulty (MLD). Because mathematical metacognition and MA predicted mathematics achievement, we compared group differences in metacognition and MA with IQ partialled out. The results showed that children with MLD scored lower in self-image and higher in learning mathematics anxiety (LMA) than the TA and HA children, but not in mathematical evaluation anxiety (MEA). MLD children’s LMA was also higher than that of their LA counterparts. These results provide insight into factors that may mediate poor WPS performance which emerges under pressure in mathematics. These results also suggest that the anxiety during learning mathematics should be taken into account in mathematical learning difficulty interventions. PMID:26090806
Koc, Ramazan . E-mail: koc@gantep.edu.tr; Tuetuencueler, Hayriye; Koca, Mehmet; Olgar, Eser
2005-10-01
We consider solutions of the 2 x 2 matrix Hamiltonians of the physical systems within the context of the su (2) and su (1, 1) Lie algebras. Our technique is relatively simple when compared with those of others and treats those Hamiltonians which can be treated in a unified framework of the Sp (4, R) algebra. The systematic study presented here reproduces a number of earlier results in a natural way as well as leads to a novel finding. Possible generalizations of the method are also suggested.
Working memory, worry, and algebraic ability.
Trezise, Kelly; Reeve, Robert A
2014-05-01
Math anxiety (MA)-working memory (WM) relationships have typically been examined in the context of arithmetic problem solving, and little research has examined the relationship in other math domains (e.g., algebra). Moreover, researchers have tended to examine MA/worry separate from math problem solving activities and have used general WM tasks rather than domain-relevant WM measures. Furthermore, it seems to have been assumed that MA affects all areas of math. It is possible, however, that MA is restricted to particular math domains. To examine these issues, the current research assessed claims about the impact on algebraic problem solving of differences in WM and algebraic worry. A sample of 80 14-year-old female students completed algebraic worry, algebraic WM, algebraic problem solving, nonverbal IQ, and general math ability tasks. Latent profile analysis of worry and WM measures identified four performance profiles (subgroups) that differed in worry level and WM capacity. Consistent with expectations, subgroup membership was associated with algebraic problem solving performance: high WM/low worry>moderate WM/low worry=moderate WM/high worry>low WM/high worry. Findings are discussed in terms of the conceptual relationship between emotion and cognition in mathematics and implications for the MA-WM-performance relationship.
ERIC Educational Resources Information Center
Jitendra, Asha K.
2011-01-01
This article presents the author's response to Yan Ping Xin and Dake Zhang's recent critical evaluation of her and colleagues' work in "Exploring a Conceptual Model-Based Approach to Teaching Situated Word Problems," published in "The Journal of Educational Research" in 2009 (Vol. 102, No. 6). Most critiques of prior research are written in a fair…
ERIC Educational Resources Information Center
Bjork, Isabel Maria; Bowyer-Crane, Claudine
2013-01-01
This study investigates the relationship between skills that underpin mathematical word problems and those that underpin numerical operations, such as addition, subtraction, division and multiplication. Sixty children aged 6-7 years were tested on measures of mathematical ability, reading accuracy, reading comprehension, verbal intelligence and…
ERIC Educational Resources Information Center
Jitendra, Asha K.; Dupuis, Danielle N.; Zaslofsky, Anne F.
2014-01-01
This purpose of this study was to examine the reliability and validity of a curriculum-based measure of word problem solving (CBM-WPS) as an indicator of performance and progress in a sample of 136 third-grade students at risk for mathematics difficulties (MDs) instructed in a standards-based mathematics curriculum. Students completed the CBM-WPS…
ERIC Educational Resources Information Center
Leh, Jayne M.; Jitendra, Asha K.; Caskie, Grace I. L.; Griffin, Cynthia C.
2007-01-01
The purpose of this study was to examine the tenability of a curriculum-based mathematical word problem-solving (WPS) measure as a progress-monitoring tool to index students' rate of growth or slope of achievement over time. Participants consisted of 58 third-grade students, who were assessed repeatedly over 16 school weeks. Students were measured…
ERIC Educational Resources Information Center
Kyttälä, Minna; Aunio, Pirjo; Lepola, Janne; Hautamäki, Jarkko
2014-01-01
The aim of this study was to analyse the role of verbal and visuo-spatial working memory (WM) and language skills (vocabulary, listening comprehension) in predicting preschool and kindergarten-aged children's ability to solve mathematical word problems presented orally. The participants were 116 Finnish-speaking children aged 4-7?years. The…
ERIC Educational Resources Information Center
Swanson, H. Lee; Moran, Amber; Lussier, Cathy; Fung, Wenson
2014-01-01
The purpose of this study was to investigate the effectiveness of explicit, direct, and generative strategy training and working memory capacity (WMC) on mathematical word problem-solving accuracy in elementary schoolchildren. In this study, children in third grade ("N" = 82) identified as at risk for math difficulties (MD) were randomly…
TBGG- INTERACTIVE ALGEBRAIC GRID GENERATION
NASA Technical Reports Server (NTRS)
Smith, R. E.
1994-01-01
TBGG, Two-Boundary Grid Generation, applies an interactive algebraic grid generation technique in two dimensions. The program incorporates mathematical equations that relate the computational domain to the physical domain. TBGG has application to a variety of problems using finite difference techniques, such as computational fluid dynamics. Examples include the creation of a C-type grid about an airfoil and a nozzle configuration in which no left or right boundaries are specified. The underlying two-boundary technique of grid generation is based on Hermite cubic interpolation between two fixed, nonintersecting boundaries. The boundaries are defined by two ordered sets of points, referred to as the top and bottom. Left and right side boundaries may also be specified, and call upon linear blending functions to conform interior interpolation to the side boundaries. Spacing between physical grid coordinates is determined as a function of boundary data and uniformly spaced computational coordinates. Control functions relating computational coordinates to parametric intermediate variables that affect the distance between grid points are embedded in the interpolation formulas. A versatile control function technique with smooth cubic spline functions is also presented. The TBGG program is written in FORTRAN 77. It works best in an interactive graphics environment where computational displays and user responses are quickly exchanged. The program has been implemented on a CDC Cyber 170 series computer using NOS 2.4 operating system, with a central memory requirement of 151,700 (octal) 60 bit words. TBGG requires a Tektronix 4015 terminal and the DI-3000 Graphics Library of Precision Visuals, Inc. TBGG was developed in 1986.
Difficulties in initial algebra learning in Indonesia
NASA Astrophysics Data System (ADS)
Jupri, Al; Drijvers, Paul; van den Heuvel-Panhuizen, Marja
2014-12-01
Within mathematics curricula, algebra has been widely recognized as one of the most difficult topics, which leads to learning difficulties worldwide. In Indonesia, algebra performance is an important issue. In the Trends in International Mathematics and Science Study (TIMSS) 2007, Indonesian students' achievement in the algebra domain was significantly below the average student performance in other Southeast Asian countries such as Thailand, Malaysia, and Singapore. This fact gave rise to this study which aims to investigate Indonesian students' difficulties in algebra. In order to do so, a literature study was carried out on students' difficulties in initial algebra. Next, an individual written test on algebra tasks was administered, followed by interviews. A sample of 51 grade VII Indonesian students worked the written test, and 37 of them were interviewed afterwards. Data analysis revealed that mathematization, i.e., the ability to translate back and forth between the world of the problem situation and the world of mathematics and to reorganize the mathematical system itself, constituted the most frequently observed difficulty in both the written test and the interview data. Other observed difficulties concerned understanding algebraic expressions, applying arithmetic operations in numerical and algebraic expressions, understanding the different meanings of the equal sign, and understanding variables. The consequences of these findings on both task design and further research in algebra education are discussed.
Algebraic methods in system theory
NASA Technical Reports Server (NTRS)
Brockett, R. W.; Willems, J. C.; Willsky, A. S.
1975-01-01
Investigations on problems of the type which arise in the control of switched electrical networks are reported. The main results concern the algebraic structure and stochastic aspects of these systems. Future reports will contain more detailed applications of these results to engineering studies.
Math for All Learners: Algebra.
ERIC Educational Resources Information Center
Meader, Pam; Storer, Judy
This book consists of a series of activities aimed at providing a problem solving, hands-on approach so that students can experience concepts in algebra. Topics include ratio and proportion, patterns and formulas, integers, polynomials, linear equations, graphs, and probability. The activities come in the form of reproducible blackline masters…
Relation of deformed nonlinear algebras with linear ones
NASA Astrophysics Data System (ADS)
Nowicki, A.; Tkachuk, V. M.
2014-01-01
The relation between nonlinear algebras and linear ones is established. For a one-dimensional nonlinear deformed Heisenberg algebra with two operators we find the function of deformation for which this nonlinear algebra can be transformed to a linear one with three operators. We also establish the relation between the Lie algebra of total angular momentum and corresponding nonlinear one. This relation gives a possibility to simplify and to solve the eigenvalue problem for the Hamiltonian in a nonlinear case using the reduction of this problem to the case of linear algebra. It is demonstrated in an example of a harmonic oscillator.
Scalable Parallel Algebraic Multigrid Solvers
Bank, R; Lu, S; Tong, C; Vassilevski, P
2005-03-23
The authors propose a parallel algebraic multilevel algorithm (AMG), which has the novel feature that the subproblem residing in each processor is defined over the entire partition domain, although the vast majority of unknowns for each subproblem are associated with the partition owned by the corresponding processor. This feature ensures that a global coarse description of the problem is contained within each of the subproblems. The advantages of this approach are that interprocessor communication is minimized in the solution process while an optimal order of convergence rate is preserved; and the speed of local subproblem solvers can be maximized using the best existing sequential algebraic solvers.
Abstract numeric relations and the visual structure of algebra.
Landy, David; Brookes, David; Smout, Ryan
2014-09-01
Formal algebras are among the most powerful and general mechanisms for expressing quantitative relational statements; yet, even university engineering students, who are relatively proficient with algebraic manipulation, struggle with and often fail to correctly deploy basic aspects of algebraic notation (Clement, 1982). In the cognitive tradition, it has often been assumed that skilled users of these formalisms treat situations in terms of semantic properties encoded in an abstract syntax that governs the use of notation without particular regard to the details of the physical structure of the equation itself (Anderson, 2005; Hegarty, Mayer, & Monk, 1995). We explore how the notational structure of verbal descriptions or algebraic equations (e.g., the spatial proximity of certain words or the visual alignment of numbers and symbols in an equation) plays a role in the process of interpreting or constructing symbolic equations. We propose in particular that construction processes involve an alignment of notational structures across representation systems, biasing reasoners toward the selection of formal notations that maintain the visuospatial structure of source representations. For example, in the statement "There are 5 elephants for every 3 rhinoceroses," the spatial proximity of 5 and elephants and 3 and rhinoceroses will bias reasoners to write the incorrect expression 5E = 3R, because that expression maintains the spatial relationships encoded in the source representation. In 3 experiments, participants constructed equations with given structure, based on story problems with a variety of phrasings. We demonstrate how the notational alignment approach accounts naturally for a variety of previously reported phenomena in equation construction and successfully predicts error patterns that are not accounted for by prior explanations, such as the left to right transcription heuristic.
ERIC Educational Resources Information Center
Ganske, Kathy
This book presents a practical approach for assessing children's spelling and word knowledge abilities and offering effective, appropriate instruction. Included in the book is the Developmental Spelling Analysis (DSA), a dictated word inventory that enables teachers to evaluate students' stages of spelling development and their knowledge of…
Algebraic sub-structuring for electromagnetic applications
Yang, Chao; Gao, Weiguo; Bai, Zhaojun; Li, Xiaoye; Lee, Lie-Quan; Husbands, Parry; Ng, Esmond G.
2004-09-14
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, we show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Algebraic Sub-Structuring for Electromagnetic Applications
Yang, C.; Gao, W.G.; Bai, Z.J.; Li, X.Y.S.; Lee, L.Q.; Husbands, P.; Ng, E.G.; /LBL, Berkeley /UC, Davis /SLAC
2006-06-30
Algebraic sub-structuring refers to the process of applying matrix reordering and partitioning algorithms to divide a large sparse matrix into smaller submatrices from which a subset of spectral components are extracted and combined to form approximate solutions to the original problem. In this paper, they show that algebraic sub-structuring can be effectively used to solve generalized eigenvalue problems arising from the finite element analysis of an accelerator structure.
Form in Algebra: Reflecting, with Peacock, on Upper Secondary School Teaching.
ERIC Educational Resources Information Center
Menghini, Marta
1994-01-01
Discusses algebra teaching by looking back into the history of algebra and the work of George Peacock, who considered algebra from two points of view: symbolic and instrumental. Claims that, to be meaningful, algebra must be linked to real-world problems. (18 references) (MKR)
Boolean Algebra. Geometry Module for Use in a Mathematics Laboratory Setting.
ERIC Educational Resources Information Center
Brotherton, Sheila; And Others
This module is recommended as an honors unit to follow a unit on logic. There are four basic parts: (1) What is a Boolean Algebra; (2) Using Boolean Algebra to Prove Theorems; (3) Using Boolean Algebra to Simplify Logical Statements; and (4) Circuit Problems with Logic and Boolean Algebra. Of these, sections 1, 2, and 3 are primarily written…
Quantization of Algebraic Reduction
Sniatycki, Jeodrzej
2007-11-14
For a Poisson algebra obtained by algebraic reduction of symmetries of a quantizable system we develop an analogue of geometric quantization based on the quantization structure of the original system.
Learning Algebra in a Computer Algebra Environment
ERIC Educational Resources Information Center
Drijvers, Paul
2004-01-01
This article summarises a doctoral thesis entitled "Learning algebra in a computer algebra environment, design research on the understanding of the concept of parameter" (Drijvers, 2003). It describes the research questions, the theoretical framework, the methodology and the results of the study. The focus of the study is on the understanding of…
Orientation in operator algebras
Alfsen, Erik M.; Shultz, Frederic W.
1998-01-01
A concept of orientation is relevant for the passage from Jordan structure to associative structure in operator algebras. The research reported in this paper bridges the approach of Connes for von Neumann algebras and ourselves for C*-algebras in a general theory of orientation that is of geometric nature and is related to dynamics. PMID:9618457
Developing Thinking in Algebra
ERIC Educational Resources Information Center
Mason, John; Graham, Alan; Johnson-Wilder, Sue
2005-01-01
This book is for people with an interest in algebra whether as a learner, or as a teacher, or perhaps as both. It is concerned with the "big ideas" of algebra and what it is to understand the process of thinking algebraically. The book has been structured according to a number of pedagogic principles that are exposed and discussed along the way,…
Connecting Arithmetic to Algebra
ERIC Educational Resources Information Center
Darley, Joy W.; Leapard, Barbara B.
2010-01-01
Algebraic thinking is a top priority in mathematics classrooms today. Because elementary school teachers lay the groundwork to develop students' capacity to think algebraically, it is crucial for teachers to have a conceptual understanding of the connections between arithmetic and algebra and be confident in communicating these connections. Many…
Computer algebra and operators
NASA Technical Reports Server (NTRS)
Fateman, Richard; Grossman, Robert
1989-01-01
The symbolic computation of operator expansions is discussed. Some of the capabilities that prove useful when performing computer algebra computations involving operators are considered. These capabilities may be broadly divided into three areas: the algebraic manipulation of expressions from the algebra generated by operators; the algebraic manipulation of the actions of the operators upon other mathematical objects; and the development of appropriate normal forms and simplification algorithms for operators and their actions. Brief descriptions are given of the computer algebra computations that arise when working with various operators and their actions.
Discrete Minimal Surface Algebras
NASA Astrophysics Data System (ADS)
Arnlind, Joakim; Hoppe, Jens
2010-05-01
We consider discrete minimal surface algebras (DMSA) as generalized noncommutative analogues of minimal surfaces in higher dimensional spheres. These algebras appear naturally in membrane theory, where sequences of their representations are used as a regularization. After showing that the defining relations of the algebra are consistent, and that one can compute a basis of the enveloping algebra, we give several explicit examples of DMSAs in terms of subsets of sln (any semi-simple Lie algebra providing a trivial example by itself). A special class of DMSAs are Yang-Mills algebras. The representation graph is introduced to study representations of DMSAs of dimension d ≤ 4, and properties of representations are related to properties of graphs. The representation graph of a tensor product is (generically) the Cartesian product of the corresponding graphs. We provide explicit examples of irreducible representations and, for coinciding eigenvalues, classify all the unitary representations of the corresponding algebras.
Chen, J.; Safro, I.
2011-01-01
Measuring the connection strength between a pair of vertices in a graph is one of the most important concerns in many graph applications. Simple measures such as edge weights may not be sufficient for capturing the effects associated with short paths of lengths greater than one. In this paper, we consider an iterative process that smooths an associated value for nearby vertices, and we present a measure of the local connection strength (called the algebraic distance; see [D. Ron, I. Safro, and A. Brandt, Multiscale Model. Simul., 9 (2011), pp. 407-423]) based on this process. The proposed measure is attractive in that the process is simple, linear, and easily parallelized. An analysis of the convergence property of the process reveals that the local neighborhoods play an important role in determining the connectivity between vertices. We demonstrate the practical effectiveness of the proposed measure through several combinatorial optimization problems on graphs and hypergraphs.
ERIC Educational Resources Information Center
Friedman, Brenda G.; And Others
Intended for language learning disabled college students and their tutors, the booklet examines the use of words as aids in understanding an author's meaning. Suggestions are offered to help students adopt a more comfortable and effective reading style. The first section examines ways to decipher authors' context cues for information. Eight…
ERIC Educational Resources Information Center
Jones, Kevin P.
1981-01-01
Proposes that the major criteria for handling compound words should rest upon their orthography (physical form), lexicography (dictionary definition), and semantics, with special attention given to possible homographs. BS 5723 is criticized for failing to pay sufficient attention to the requirements of mechanized systems. Thirty-one references are…
A Richer Understanding of Algebra
ERIC Educational Resources Information Center
Foy, Michelle
2008-01-01
Algebra is one of those hard-to-teach topics where pupils seem to struggle to see it as more than a set of rules to learn, but this author recently used the software "Grid Algebra" from ATM, which engaged her Year 7 pupils in exploring algebraic concepts for themselves. "Grid Algebra" allows pupils to experience number, pre-algebra, and algebra…
What Is the Place of Algebra in the K-12 Mathematics Program?
ERIC Educational Resources Information Center
Fendel, Dan; And Others
1997-01-01
As times change, so has the role of algebra in the educational program. The Interactive Mathematics Program (IMP) offers secondary students an opportunity to learn algebra in a college preparatory sequence that combines basic skills, problem solving, and conceptual understanding while integrating algebra into a problem-based program. Designed for…
Connecting Algebra and Chemistry.
ERIC Educational Resources Information Center
O'Connor, Sean
2003-01-01
Correlates high school chemistry curriculum with high school algebra curriculum and makes the case for an integrated approach to mathematics and science instruction. Focuses on process integration. (DDR)
Vihman, M M; McCune, L
1994-10-01
Although adult-based words co-occur in the period of transition to speech with a variety of non-word vocalizations, little attention has been given to the formidable problem of identifying these earliest words. This paper specifies explicit, maximally 'inclusive' identification procedures, with criteria based on both phonetic and contextual parameters. A formal system for evaluating phonetic match is suggested, as well as a set of child-derived functional categories reflecting use in context. Analysis of word use across two samples of 10 children each, followed from 0;9 to 1;4, provides evidence to suggest that context-bound words can be 'trained' by focusing on eliciting language, but that the timing of context-flexible word use remains independent of such training.
A Linear Algebraic Approach to Teaching Interpolation
ERIC Educational Resources Information Center
Tassa, Tamir
2007-01-01
A novel approach for teaching interpolation in the introductory course in numerical analysis is presented. The interpolation problem is viewed as a problem in linear algebra, whence the various forms of interpolating polynomial are seen as different choices of a basis to the subspace of polynomials of the corresponding degree. This approach…
Digital Maps, Matrices and Computer Algebra
ERIC Educational Resources Information Center
Knight, D. G.
2005-01-01
The way in which computer algebra systems, such as Maple, have made the study of complex problems accessible to undergraduate mathematicians with modest computational skills is illustrated by some large matrix calculations, which arise from representing the Earth's surface by digital elevation models. Such problems are often considered to lie in…
Automated Angular Momentum Recoupling Algebra
NASA Astrophysics Data System (ADS)
Williams, H. T.; Silbar, Richard R.
1992-04-01
We present a set of heuristic rules for algebraic solution of angular momentum recoupling problems. The general problem reduces to that of finding an optimal path from one binary tree (representing the angular momentum coupling scheme for the reduced matrix element) to another (representing the sub-integrals and spin sums to be done). The method lends itself to implementation on a microcomputer, and we have developed such an implementation using a dialect of LISP. We describe both how our code, called RACAH, works and how it appears to the user. We illustrate the use of RACAH for several transition and scattering amplitude matrix elements occurring in atomic, nuclear, and particle physics.
NASA Technical Reports Server (NTRS)
Lawson, C. L.; Krogh, F. T.; Gold, S. S.; Kincaid, D. R.; Sullivan, J.; Williams, E.; Hanson, R. J.; Haskell, K.; Dongarra, J.; Moler, C. B.
1982-01-01
The Basic Linear Algebra Subprograms (BLAS) library is a collection of 38 FORTRAN-callable routines for performing basic operations of numerical linear algebra. BLAS library is portable and efficient source of basic operations for designers of programs involving linear algebriac computations. BLAS library is supplied in portable FORTRAN and Assembler code versions for IBM 370, UNIVAC 1100 and CDC 6000 series computers.
ERIC Educational Resources Information Center
Levy, Alissa Beth
2012-01-01
The California Department of Education (CDE) has long asserted that success Algebra I by Grade 8 is the goal for all California public school students. In fact, the state's accountability system penalizes schools that do not require all of their students to take the Algebra I end-of-course examination by Grade 8 (CDE, 2009). In this…
Algebraic Reasoning through Patterns
ERIC Educational Resources Information Center
Rivera, F. D.; Becker, Joanne Rossi
2009-01-01
This article presents the results of a three-year study that explores students' performance on patterning tasks involving prealgebra and algebra. The findings, insights, and issues drawn from the study are intended to help teach prealgebra and algebra. In the remainder of the article, the authors take a more global view of the three-year study on…
ERIC Educational Resources Information Center
Merlin, Ethan M.
2013-01-01
This article describes how the author has developed tasks for students that address the missed "essence of the matter" of algebraic transformations. Specifically, he has found that having students practice "perceiving" algebraic structure--by naming the "glue" in the expressions, drawing expressions using…
Lefrancois, Daniel; Wormit, Michael; Dreuw, Andreas
2015-09-28
For the investigation of molecular systems with electronic ground states exhibiting multi-reference character, a spin-flip (SF) version of the algebraic diagrammatic construction (ADC) scheme for the polarization propagator up to third order perturbation theory (SF-ADC(3)) is derived via the intermediate state representation and implemented into our existing ADC computer program adcman. The accuracy of these new SF-ADC(n) approaches is tested on typical situations, in which the ground state acquires multi-reference character, like bond breaking of H{sub 2} and HF, the torsional motion of ethylene, and the excited states of rectangular and square-planar cyclobutadiene. Overall, the results of SF-ADC(n) reveal an accurate description of these systems in comparison with standard multi-reference methods. Thus, the spin-flip versions of ADC are easy-to-use methods for the calculation of “few-reference” systems, which possess a stable single-reference triplet ground state.
Promoting Quantitative Literacy in an Online College Algebra Course
ERIC Educational Resources Information Center
Tunstall, Luke; Bossé, Michael J.
2016-01-01
College algebra (a university freshman level algebra course) fulfills the quantitative literacy requirement of many college's general education programs and is a terminal course for most who take it. An online problem-based learning environment provides a unique means of engaging students in quantitative discussions and research. This article…
Algebraic Concepts: What's Really New in New Curricula?
ERIC Educational Resources Information Center
Star, Jon R.; Herbel-Eisenmann, Beth A.; Smith, John P., III
2000-01-01
Examines 8th grade units from the Connected Mathematics Project (CMP). Identifies differences in older and newer conceptions, fundamental objects of study, typical problems, and typical solution methods in algebra. Also discusses where the issue of what is new in algebra is relevant to many other innovative middle school curricula. (KHR)
Lie algebra extensions of current algebras on S3
NASA Astrophysics Data System (ADS)
Kori, Tosiaki; Imai, Yuto
2015-06-01
An affine Kac-Moody algebra is a central extension of the Lie algebra of smooth mappings from S1 to the complexification of a Lie algebra. In this paper, we shall introduce a central extension of the Lie algebra of smooth mappings from S3 to the quaternization of a Lie algebra and investigate its root space decomposition. We think this extension of current algebra might give a mathematical tool for four-dimensional conformal field theory as Kac-Moody algebras give it for two-dimensional conformal field theory.
ERIC Educational Resources Information Center
Clemson Univ., SC. Vocational Education Media Center.
Designed for use in teaching secondary-level word processing courses, this teaching guide is divided into three major sections. Among the topics presented in the introductory section are the history of word processing, components of word processing, five phases of word processing, the future of word processing and information systems, and job…
Computer algebra and transport theory.
Warsa, J. S.
2004-01-01
Modern symbolic algebra computer software augments and complements more traditional approaches to transport theory applications in several ways. The first area is in the development and enhancement of numerical solution methods for solving the Boltzmann transport equation. Typically, special purpose computer codes are designed and written to solve specific transport problems in particular ways. Different aspects of the code are often written from scratch and the pitfalls of developing complex computer codes are numerous and well known. Software such as MAPLE and MATLAB can be used to prototype, analyze, verify and determine the suitability of numerical solution methods before a full-scale transport application is written. Once it is written, the relevant pieces of the full-scale code can be verified using the same tools I that were developed for prototyping. Another area is in the analysis of numerical solution methods or the calculation of theoretical results that might otherwise be difficult or intractable. Algebraic manipulations are done easily and without error and the software also provides a framework for any additional numerical calculations that might be needed to complete the analysis. We will discuss several applications in which we have extensively used MAPLE and MATLAB in our work. All of them involve numerical solutions of the S{sub N} transport equation. These applications encompass both of the two main areas in which we have found computer algebra software essential.
Leibniz algebras associated with representations of filiform Lie algebras
NASA Astrophysics Data System (ADS)
Ayupov, Sh. A.; Camacho, L. M.; Khudoyberdiyev, A. Kh.; Omirov, B. A.
2015-12-01
In this paper we investigate Leibniz algebras whose quotient Lie algebra is a naturally graded filiform Lie algebra nn,1. We introduce a Fock module for the algebra nn,1 and provide classification of Leibniz algebras L whose corresponding Lie algebra L / I is the algebra nn,1 with condition that the ideal I is a Fock nn,1-module, where I is the ideal generated by squares of elements from L. We also consider Leibniz algebras with corresponding Lie algebra nn,1 and such that the action I ×nn,1 → I gives rise to a minimal faithful representation of nn,1. The classification up to isomorphism of such Leibniz algebras is given for the case of n = 4.
Optical systolic solutions of linear algebraic equations
NASA Technical Reports Server (NTRS)
Neuman, C. P.; Casasent, D.
1984-01-01
The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.
NASA Astrophysics Data System (ADS)
Smirnov, Andrey
2010-08-01
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl( N;?)-case is discussed.
NASA Astrophysics Data System (ADS)
Smirnov, Andrey
2010-08-01
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for sl(2;?). These solutions are shown to be related to the standard ones by the quasi-Hopf twist. We demonstrate that the quantum algebras arising from these new R-matrices can be obtained as special limits of the Sklyanin algebra. A representation for these algebras by the difference operators is found. The sl(N;?)-case is discussed.
The Taylor spectrum and transversality for a Heisenberg algebra of operators
NASA Astrophysics Data System (ADS)
Dosi, Anar A.
2010-05-01
A problem on noncommutative holomorphic functional calculus is considered for a Banach module over a finite-dimensional nilpotent Lie algebra. As the main result, the transversality property of algebras of noncommutative holomorphic functions with respect to the Taylor spectrum is established for a family of bounded linear operators generating a Heisenberg algebra. Bibliography: 25 titles.
The Taylor spectrum and transversality for a Heisenberg algebra of operators
Dosi, Anar A
2010-05-11
A problem on noncommutative holomorphic functional calculus is considered for a Banach module over a finite-dimensional nilpotent Lie algebra. As the main result, the transversality property of algebras of noncommutative holomorphic functions with respect to the Taylor spectrum is established for a family of bounded linear operators generating a Heisenberg algebra. Bibliography: 25 titles.
Algebraic integrability: a survey.
Vanhaecke, Pol
2008-03-28
We give a concise introduction to the notion of algebraic integrability. Our exposition is based on examples and phenomena, rather than on detailed proofs of abstract theorems. We mainly focus on algebraic integrability in the sense of Adler-van Moerbeke, where the fibres of the momentum map are affine parts of Abelian varieties; as it turns out, most examples from classical mechanics are of this form. Two criteria are given for such systems (Kowalevski-Painlevé and Lyapunov) and each is illustrated in one example. We show in the case of a relatively simple example how one proves algebraic integrability, starting from the differential equations for the integrable vector field. For Hamiltonian systems that are algebraically integrable in the generalized sense, two examples are given, which illustrate the non-compact analogues of Abelian varieties which typically appear in such systems. PMID:17588863
Algebraic Semantics for Narrative
ERIC Educational Resources Information Center
Kahn, E.
1974-01-01
This paper uses discussion of Edmund Spenser's "The Faerie Queene" to present a theoretical framework for explaining the semantics of narrative discourse. The algebraic theory of finite automata is used. (CK)
Covariant deformed oscillator algebras
NASA Technical Reports Server (NTRS)
Quesne, Christiane
1995-01-01
The general form and associativity conditions of deformed oscillator algebras are reviewed. It is shown how the latter can be fulfilled in terms of a solution of the Yang-Baxter equation when this solution has three distinct eigenvalues and satisfies a Birman-Wenzl-Murakami condition. As an example, an SU(sub q)(n) x SU(sub q)(m)-covariant q-bosonic algebra is discussed in some detail.
Aprepro - Algebraic Preprocessor
2005-08-01
Aprepro is an algebraic preprocessor that reads a file containing both general text and algebraic, string, or conditional expressions. It interprets the expressions and outputs them to the output file along witht the general text. Aprepro contains several mathematical functions, string functions, and flow control constructs. In addition, functions are included that, with some additional files, implement a units conversion system and a material database lookup system.
Geometric Algebra for Physicists
NASA Astrophysics Data System (ADS)
Doran, Chris; Lasenby, Anthony
2007-11-01
Preface; Notation; 1. Introduction; 2. Geometric algebra in two and three dimensions; 3. Classical mechanics; 4. Foundations of geometric algebra; 5. Relativity and spacetime; 6. Geometric calculus; 7. Classical electrodynamics; 8. Quantum theory and spinors; 9. Multiparticle states and quantum entanglement; 10. Geometry; 11. Further topics in calculus and group theory; 12. Lagrangian and Hamiltonian techniques; 13. Symmetry and gauge theory; 14. Gravitation; Bibliography; Index.
NASA Astrophysics Data System (ADS)
Hiley, B. J.
In this chapter, we examine in detail the non-commutative symplectic algebra underlying quantum dynamics. By using this algebra, we show that it contains both the Weyl-von Neumann and the Moyal quantum algebras. The latter contains the Wigner distribution as the kernel of the density matrix. The underlying non-commutative geometry can be projected into either of two Abelian spaces, so-called `shadow phase spaces'. One of these is the phase space of Bohmian mechanics, showing that it is a fragment of the basic underlying algebra. The algebraic approach is much richer, giving rise to two fundamental dynamical time development equations which reduce to the Liouville equation and the Hamilton-Jacobi equation in the classical limit. They also include the Schrödinger equation and its wave-function, showing that these features are a partial aspect of the more general non-commutative structure. We discuss briefly the properties of this more general mathematical background from which the non-commutative symplectic algebra emerges.
Stability of Linear Equations--Algebraic Approach
ERIC Educational Resources Information Center
Cherif, Chokri; Goldstein, Avraham; Prado, Lucio M. G.
2012-01-01
This article could be of interest to teachers of applied mathematics as well as to people who are interested in applications of linear algebra. We give a comprehensive study of linear systems from an application point of view. Specifically, we give an overview of linear systems and problems that can occur with the computed solution when the…
Parallel Algebraic Multigrids for Structural mechanics
Brezina, M; Tong, C; Becker, R
2004-05-11
This paper presents the results of a comparison of three parallel algebraic multigrid (AMG) preconditioners for structural mechanics applications. In particular, they are interested in investigating both the scalability and robustness of the preconditioners. Numerical results are given for a range of structural mechanics problems with various degrees of difficulty.
Inequalities, assessment and computer algebra
NASA Astrophysics Data System (ADS)
Sangwin, Christopher J.
2015-01-01
The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students' own answers to such problems are correct. We review how inequalities arise in contemporary curricula. We consider the formal mathematical processes by which such inequalities are solved, and we consider the notation and syntax through which solutions are expressed. We review the extent to which current CAS can accurately solve these inequalities, and the form given to the solutions by the designers of this software. Finally, we discuss the functionality needed to deal with students' answers, i.e. to establish equivalence (or otherwise) of expressions representing unions of intervals. We find that while contemporary CAS accurately solve inequalities there is a wide variety of notation used.
Local Algebras of Differential Operators
NASA Astrophysics Data System (ADS)
Church, P. T.; Timourian, J. G.
2002-05-01
There is an increasing literature devoted to the study of boundary value problems using singularity theory. The resulting differential operators are typically Fredholm with index 0, defined on infinite-dimensional spaces, and they have often led to folds, cusps, and even higher-order Morin singularities. In this paper we develop some of the local algebras of germs of such differential Fredholm operators, extending the theory of the finite-dimensional case. We apply this work to nonlinear elliptic boundary value problems: in particular, we make further progress on a question proposed and initially studied by Ruf [1999, J. Differential Equations 151, 111-133]. We also make comments on several problems raised by others.
SLAPP: A systolic linear algebra parallel processor
Drake, B.L.; Luk, F.T.; Speiser, J.M.; Symanski, J.J.
1987-07-01
Systolic array computer architectures provide a means for fast computation of the linear algebra algorithms that form the building blocks of many signal-processing algorithms, facilitating their real-time computation. For applications to signal processing, the systolic array operates on matrices, an inherently parallel view of the data, using numerical linear algebra algorithms that have been suitably parallelized to efficiently utilize the available hardware. This article describes work currently underway at the Naval Ocean Systems Center, San Diego, California, to build a two-dimensional systolic array, SLAPP, demonstrating efficient and modular parallelization of key matric computations for real-time signal- and image-processing problems.
Effective mass Schrödinger equation and nonlinear algebras
NASA Astrophysics Data System (ADS)
Roy, B.; Roy, P.
2005-06-01
Using supersymmetry we obtain solutions of Schrödinger equation with a position dependent effective mass exhibiting a harmonic oscillator like spectrum. We also discuss the underlying nonlinear algebraic symmetry of the problem.
Applied Algebra: The Modeling Technique of Least Squares
ERIC Educational Resources Information Center
Zelkowski, Jeremy; Mayes, Robert
2008-01-01
The article focuses on engaging students in algebra through modeling real-world problems. The technique of least squares is explored, encouraging students to develop a deeper understanding of the method. (Contains 2 figures and a bibliography.)
ERIC Educational Resources Information Center
Swanson, H. Lee; Orosco, Michael J.; Lussier, Cathy
2013-01-01
Recent intervention studies directed to improve problem solving accuracy in children with math difficulties (MD) have found support for teaching cognitive strategies. This study addresses the question: What role does working memory capacity (WMC) play in strategy outcomes for children with MD? Four prediction models can be applied to strategy…
Abstract Algebra for Algebra Teaching: Influencing School Mathematics Instruction
ERIC Educational Resources Information Center
Wasserman, Nicholas H.
2016-01-01
This article explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics--and their progression across elementary, middle, and secondary mathematics--where teaching may be transformed by…
q-graded Heisenberg algebras and deformed supersymmetries
Ben Geloun, Joseph; Hounkonnou, Mahouton Norbert
2010-02-15
The notion of q-grading on the enveloping algebra generated by products of q-deformed Heisenberg algebras is introduced for q complex number in the unit disk. Within this formulation, we consider the extension of the notion of supersymmetry in the enveloping algebra. We recover the ordinary Z{sub 2} grading or Grassmann parity for associative superalgebra and a modified version of the usual supersymmetry. As a specific problem, we focus on the interesting limit q{yields}-1 for which the Arik and Coon deformation [J. Math. Phys. 17, 524 (1976)] of the Heisenberg algebra allows one to map fermionic modes to bosonic ones in a modified sense. Different algebraic consequences are discussed.
Adaptive Algebraic Multigrid Methods
Brezina, M; Falgout, R; MacLachlan, S; Manteuffel, T; McCormick, S; Ruge, J
2004-04-09
Our ability to simulate physical processes numerically is constrained by our ability to solve the resulting linear systems, prompting substantial research into the development of multiscale iterative methods capable of solving these linear systems with an optimal amount of effort. Overcoming the limitations of geometric multigrid methods to simple geometries and differential equations, algebraic multigrid methods construct the multigrid hierarchy based only on the given matrix. While this allows for efficient black-box solution of the linear systems associated with discretizations of many elliptic differential equations, it also results in a lack of robustness due to assumptions made on the near-null spaces of these matrices. This paper introduces an extension to algebraic multigrid methods that removes the need to make such assumptions by utilizing an adaptive process. The principles which guide the adaptivity are highlighted, as well as their application to algebraic multigrid solution of certain symmetric positive-definite linear systems.
Computer Program For Linear Algebra
NASA Technical Reports Server (NTRS)
Krogh, F. T.; Hanson, R. J.
1987-01-01
Collection of routines provided for basic vector operations. Basic Linear Algebra Subprogram (BLAS) library is collection from FORTRAN-callable routines for employing standard techniques to perform basic operations of numerical linear algebra.
Algebra for Gifted Third Graders.
ERIC Educational Resources Information Center
Borenson, Henry
1987-01-01
Elementary school children who are exposed to a concrete, hands-on experience in algebraic linear equations will more readily develop a positive mind-set and expectation for success in later formal, algebraic studies. (CB)
Assessing Elementary Algebra with STACK
ERIC Educational Resources Information Center
Sangwin, Christopher J.
2007-01-01
This paper concerns computer aided assessment (CAA) of mathematics in which a computer algebra system (CAS) is used to help assess students' responses to elementary algebra questions. Using a methodology of documentary analysis, we examine what is taught in elementary algebra. The STACK CAA system, http://www.stack.bham.ac.uk/, which uses the CAS…
Situating the Debate on "Geometrical Algebra" within the Framework of Premodern Algebra.
Sialaros, Michalis; Christianidis, Jean
2016-06-01
Argument The aim of this paper is to employ the newly contextualized historiographical category of "premodern algebra" in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on "geometrical algebra." Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called "semi-algebraic" alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing "premodern algebra," and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition. PMID:27171890
Situating the Debate on "Geometrical Algebra" within the Framework of Premodern Algebra.
Sialaros, Michalis; Christianidis, Jean
2016-06-01
Argument The aim of this paper is to employ the newly contextualized historiographical category of "premodern algebra" in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on "geometrical algebra." Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related to Elem. II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called "semi-algebraic" alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing "premodern algebra," and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.
LAPACK: Linear algebra software for supercomputers
Bischof, C.H.
1991-01-01
This paper presents an overview of the LAPACK library, a portable, public-domain library to solve the most common linear algebra problems. This library provides a uniformly designed set of subroutines for solving systems of simultaneous linear equations, least-squares problems, and eigenvalue problems for dense and banded matrices. We elaborate on the design methodologies incorporated to make the LAPACK codes efficient on today's high-performance architectures. In particular, we discuss the use of block algorithms and the reliance on the Basic Linear Algebra Subprograms. We present performance results that show the suitability of the LAPACK approach for vector uniprocessors and shared-memory multiprocessors. We also address some issues that have to be dealt with in tuning LAPACK for specific architectures. Lastly, we present results that show that the LAPACK software can be adapted with little effort to distributed-memory environments, and we discuss future efforts resulting from this project. 31 refs., 10 figs., 2 tabs.
Factors influencing the algebra ``reversal error''
NASA Astrophysics Data System (ADS)
Cohen, Elaine; Kanim, Stephen E.
2005-11-01
Given a written problem statement about a proportional relationship between two quantities, many students will place the constant of proportionality on the wrong side of the equals sign. Introductory physics is one of the first courses in which students encounter multiple-step problems that require algebraic (rather than numeric) solutions, and this "reversal error" is relatively common in student solutions to these types of problems. We describe an investigation into three possible influences on students who make this reversal error: variable symbol choice, sentence structure, and context familiarity. Our results, from a calculus-based physics course and an intermediate algebra course, show that sentence structure is the most significant of these three possibilities. However, sentence structure alone does not provide a complete explanation for the reversal error.
... Signal Words? Signal words are found on pesticide product labels, and they describe the acute (short-term) toxicity ... red letters on the front panel of the product label. 2,4 Acute Oral LD 50 Inhalation LC ...
Trofatter, Caroline; Kontra, Carly; Beilock, Sian; Goldin-Meadow, Susan
2014-01-01
The coordination of speech with gesture elicits changes in speakers’ problem-solving behavior beyond the changes elicited by the coordination of speech with action. Participants solved the Tower of Hanoi puzzle (TOH1); explained their solution using speech coordinated with either Gestures (Gesture+Talk) or Actions (Action+Talk), or demonstrated their solution using Actions alone (Action); then solved the puzzle again (TOH2). For some participants (Switch group), disk weights during TOH2 were reversed (smallest = heaviest). Only in the Gesture+Talk Switch group did performance worsen from TOH1 to TOH2 – for all other groups, performance improved. In the Gesture+Talk Switch group, more one-handed gestures about the smallest disk during the explanation hurt subsequent performance, compared to all other groups. These findings contradict the hypothesis that gesture affects thought by promoting the coordination of task-relevant hand movements with task-relevant speech, and lend support to the hypothesis that gesture grounds thought in action via its representational properties. PMID:25664327
ERIC Educational Resources Information Center
Leh, Jayne
2011-01-01
Substantial evidence indicates that teacher-delivered schema-based instruction (SBI) facilitates significant increases in mathematics word problem solving (WPS) skills for diverse students; however research is unclear whether technology affordances facilitate superior gains in computer-mediated (CM) instruction in mathematics WPS when compared to…
Fibonacci's Triangle: A Vehicle for Problem Solving.
ERIC Educational Resources Information Center
Ouellette, Hugh
1979-01-01
A method for solving certain types of problems is illustrated by problems related to Fibonacci's triangle. The method involves pattern recognition, generalizing, algebraic manipulation, and mathematical induction. (MP)
ERIC Educational Resources Information Center
Benjamin, Carl; And Others
Presented are student performance objectives, a student progress chart, and assignment sheets with objective and diagnostic measures for the stated performance objectives in College Algebra II. Topics covered include: differencing and complements; real numbers; factoring; fractions; linear equations; exponents and radicals; complex numbers,…
Thinking Visually about Algebra
ERIC Educational Resources Information Center
Baroudi, Ziad
2015-01-01
Many introductions to algebra in high school begin with teaching students to generalise linear numerical patterns. This article argues that this approach needs to be changed so that students encounter variables in the context of modelling visual patterns so that the variables have a meaning. The article presents sample classroom activities,…
ERIC Educational Resources Information Center
Nwabueze, Kenneth K.
2004-01-01
The current emphasis on flexible modes of mathematics delivery involving new information and communication technology (ICT) at the university level is perhaps a reaction to the recent change in the objectives of education. Abstract algebra seems to be one area of mathematics virtually crying out for computer instructional support because of the…
Mitter conjecture and structure theorem for six-dimensional estimation algebras
NASA Astrophysics Data System (ADS)
Jiao, Yang; Yau, Stephen; Chiou, Wen-Lin
2013-01-01
The problem of classification of finite-dimensional estimation algebras was formally proposed by Brockett in his lecture at International Congress of Mathematicians in 1983. Due to the difficulty of the problem, in the early 1990s Brockett suggested that one should understand the low-dimensional estimation algebras first. In this article, we extend Yau and his coauthors' work of the Mitter conjecture for low-dimensional estimation algebras in nonlinear filtering problem. And, we apply the results to give classification of estimation algebras of dimension six.
Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme
NASA Astrophysics Data System (ADS)
Mazzocco, Marta
2016-09-01
In this paper we produce seven new algebras as confluences of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877–912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory.
Confluences of the Painlevé equations, Cherednik algebras and q-Askey scheme
NASA Astrophysics Data System (ADS)
Mazzocco, Marta
2016-09-01
In this paper we produce seven new algebras as confluences of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type \\check {{{{C}1}}} {{C}1} is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877-912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory.
A Schwinger Term in q-Deformed su(2) Algebra
NASA Astrophysics Data System (ADS)
Fujikawa, Kazuo; Kubo, Harunobu; Oh, C. H.
An extra term generally appears in the q-deformed su(2) algebra for the deformation parameter q=exp2π iθ, if one combines the Biedenharn-Macfarlane construction of q-deformed su(2), which is a generalization of Schwinger's construction of conventional su(2), with the representation of the q-deformed oscillator algebra which is manifestly free of negative norm. This extra term introduced by the requirement of positive norm is analogous to the Schwinger term in current algebra. Implications of this extra term on the Bloch electron problem analyzed by Wiegmann and Zabrodin are briefly discussed.
Deformable target tracking method based on Lie algebra
NASA Astrophysics Data System (ADS)
Liu, Yunpeng; Shi, Zelin; Li, Guangwei
2007-11-01
Conventional approaches to object tracking use area correlation, but they are difficult to solve the problem of deformation of object region during tracking. A novel target tracking method based on Lie algebra is presented. We use Gabor feature as target token, model deformation using affine Lie group, and optimize parameters directly on manifold, which can be solved by exponential mapping between Lie Group and its Lie algebra. We analyze the essence of our method and test the algorithm using real image sequences. The experimental results demonstrate that Lie algebra method outperforms other traditional algorithms in efficiency, stabilization and accuracy.
Algebraic connectivity and graph robustness.
Feddema, John Todd; Byrne, Raymond Harry; Abdallah, Chaouki T.
2009-07-01
Recent papers have used Fiedler's definition of algebraic connectivity to show that network robustness, as measured by node-connectivity and edge-connectivity, can be increased by increasing the algebraic connectivity of the network. By the definition of algebraic connectivity, the second smallest eigenvalue of the graph Laplacian is a lower bound on the node-connectivity. In this paper we show that for circular random lattice graphs and mesh graphs algebraic connectivity is a conservative lower bound, and that increases in algebraic connectivity actually correspond to a decrease in node-connectivity. This means that the networks are actually less robust with respect to node-connectivity as the algebraic connectivity increases. However, an increase in algebraic connectivity seems to correlate well with a decrease in the characteristic path length of these networks - which would result in quicker communication through the network. Applications of these results are then discussed for perimeter security.
Numerical solution to systems of delay integrodifferential algebraic equations
NASA Astrophysics Data System (ADS)
Dmitriev, S. S.; Kuznetsov, E. B.
2008-03-01
The numerical solution of the initial value problem for a system of delay integrodifferential algebraic equations is examined in the framework of the parametric continuation method. Necessary and sufficient conditions are obtained for transforming this problem to the best argument, which is the arc length along the integral curve of the problem. The efficiency of the transformation is demonstrated using test examples.
Maximizing algebraic connectivity in air transportation networks
NASA Astrophysics Data System (ADS)
Wei, Peng
In air transportation networks the robustness of a network regarding node and link failures is a key factor for its design. An experiment based on the real air transportation network is performed to show that the algebraic connectivity is a good measure for network robustness. Three optimization problems of algebraic connectivity maximization are then formulated in order to find the most robust network design under different constraints. The algebraic connectivity maximization problem with flight routes addition or deletion is first formulated. Three methods to optimize and analyze the network algebraic connectivity are proposed. The Modified Greedy Perturbation Algorithm (MGP) provides a sub-optimal solution in a fast iterative manner. The Weighted Tabu Search (WTS) is designed to offer a near optimal solution with longer running time. The relaxed semi-definite programming (SDP) is used to set a performance upper bound and three rounding techniques are discussed to find the feasible solution. The simulation results present the trade-off among the three methods. The case study on two air transportation networks of Virgin America and Southwest Airlines show that the developed methods can be applied in real world large scale networks. The algebraic connectivity maximization problem is extended by adding the leg number constraint, which considers the traveler's tolerance for the total connecting stops. The Binary Semi-Definite Programming (BSDP) with cutting plane method provides the optimal solution. The tabu search and 2-opt search heuristics can find the optimal solution in small scale networks and the near optimal solution in large scale networks. The third algebraic connectivity maximization problem with operating cost constraint is formulated. When the total operating cost budget is given, the number of the edges to be added is not fixed. Each edge weight needs to be calculated instead of being pre-determined. It is illustrated that the edge addition and the
Algebraic criteria for positive realness relative to the unit circle.
NASA Technical Reports Server (NTRS)
Siljak, D. D.
1973-01-01
A definition is presented of the circle positive realness of real rational functions relative to the unit circle in the complex variable plane. The problem of testing this kind of positive reality is reduced to the algebraic problem of determining the distribution of zeros of a real polynomial with respect to and on the unit circle. Such reformulation of the problem avoids the search for explicit information about imaginary poles of rational functions. The stated algebraic problem is solved by applying the polynomial criteria of Marden (1966) and Jury (1964), and a completely recursive algorithm for circle positive realness is obtained.
Algebraic curves of maximal cyclicity
NASA Astrophysics Data System (ADS)
Caubergh, Magdalena; Dumortier, Freddy
2006-01-01
The paper deals with analytic families of planar vector fields, studying methods to detect the cyclicity of a non-isolated closed orbit, i.e. the maximum number of limit cycles that can locally bifurcate from it. It is known that this multi-parameter problem can be reduced to a single-parameter one, in the sense that there exist analytic curves in parameter space along which the maximal cyclicity can be attained. In that case one speaks about a maximal cyclicity curve (mcc) in case only the number is considered and of a maximal multiplicity curve (mmc) in case the multiplicity is also taken into account. In view of obtaining efficient algorithms for detecting the cyclicity, we investigate whether such mcc or mmc can be algebraic or even linear depending on certain general properties of the families or of their associated Bautin ideal. In any case by well chosen examples we show that prudence is appropriate.
PC Basic Linear Algebra Subroutines
1992-03-09
PC-BLAS is a highly optimized version of the Basic Linear Algebra Subprograms (BLAS), a standardized set of thirty-eight routines that perform low-level operations on vectors of numbers in single and double-precision real and complex arithmetic. Routines are included to find the index of the largest component of a vector, apply a Givens or modified Givens rotation, multiply a vector by a constant, determine the Euclidean length, perform a dot product, swap and copy vectors, andmore » find the norm of a vector. The BLAS have been carefully written to minimize numerical problems such as loss of precision and underflow and are designed so that the computation is independent of the interface with the calling program. This independence is achieved through judicious use of Assembly language macros. Interfaces are provided for Lahey Fortran 77, Microsoft Fortran 77, and Ryan-McFarland IBM Professional Fortran.« less
Linear Algebraic Method for Non-Linear Map Analysis
Yu,L.; Nash, B.
2009-05-04
We present a newly developed method to analyze some non-linear dynamics problems such as the Henon map using a matrix analysis method from linear algebra. Choosing the Henon map as an example, we analyze the spectral structure, the tune-amplitude dependence, the variation of tune and amplitude during the particle motion, etc., using the method of Jordan decomposition which is widely used in conventional linear algebra.
NASA Astrophysics Data System (ADS)
Clemmons, Karina
Vocabulary in a second language is an indispensable building block of all comprehension (Folse, 2006; Nation, 2006). Teachers in content area classes such as science, math, and social studies frequently teach content specific vocabulary, but are not aware of the obstacles that can occur when students do not know the basic words. Word lists such as the General Service List (GSL) were created to assist students and teachers (West, 1953). The GSL does not adequately take into account the high level of polysemy of many common English words, nor has it been updated by genre to reflect specific content domains encountered by secondary science students in today's high stakes classes such as chemistry. This study examines how many words of the first 1000 words of the GSL occurred in the secondary chemistry textbooks sampled, how often the first 1000 words of the GSL were polysemous, and specifically which multiple meanings occurred. A discussion of results includes word tables that list multiple meanings present, example phrases that illustrate the context surrounding the target words, suggestions for a GSL that is genre specific to secondary chemistry textbooks and that is ranked by meaning as well as type, and implications for both vocabulary materials and classroom instruction for ELLs in secondary chemistry classes. Findings are essential to second language (L2) researchers, materials developers, publishers, and teachers.
Selecting reusable components using algebraic specifications
NASA Technical Reports Server (NTRS)
Eichmann, David A.
1992-01-01
A significant hurdle confronts the software reuser attempting to select candidate components from a software repository - discriminating between those components without resorting to inspection of the implementation(s). We outline a mixed classification/axiomatic approach to this problem based upon our lattice-based faceted classification technique and Guttag and Horning's algebraic specification techniques. This approach selects candidates by natural language-derived classification, by their interfaces, using signatures, and by their behavior, using axioms. We briefly outline our problem domain and related work. Lattice-based faceted classifications are described; the reader is referred to surveys of the extensive literature for algebraic specification techniques. Behavioral support for reuse queries is presented, followed by the conclusions.
Word form Encoding in Chinese Word Naming and Word Typing
ERIC Educational Resources Information Center
Chen, Jenn-Yeu; Li, Cheng-Yi
2011-01-01
The process of word form encoding was investigated in primed word naming and word typing with Chinese monosyllabic words. The target words shared or did not share the onset consonants with the prime words. The stimulus onset asynchrony (SOA) was 100 ms or 300 ms. Typing required the participants to enter the phonetic letters of the target word,…
NASA Astrophysics Data System (ADS)
Dankova, T. S.; Rosensteel, G.
1998-10-01
Mean field theory has an unexpected group theoretic mathematical foundation. Instead of representation theory which applies to most group theoretic quantum models, Hartree-Fock and Hartree-Fock-Bogoliubov have been formulated in terms of coadjoint orbits for the groups U(n) and O(2n). The general theory of mean fields is formulated for an arbitrary Lie algebra L of fermion operators. The moment map provides the correspondence between the Hilbert space of microscopic wave functions and the dual space L^* of densities. The coadjoint orbits of the group in the dual space are phase spaces on which time-dependent mean field theory is equivalent to a classical Hamiltonian dynamical system. Indeed it forms a finite-dimensional Lax system. The mean field theories for the Elliott SU(3) and symplectic Sp(3,R) algebras are constructed explicitly in the coadjoint orbit framework.
ERIC Educational Resources Information Center
Beigie, Darin
2014-01-01
Most people who are attracted to STEM-related fields are drawn not by a desire to take mathematics tests but to create things. The opportunity to create an algebra drawing gives students a sense of ownership and adventure that taps into the same sort of energy that leads a young person to get lost in reading a good book, building with Legos®,…
NASA Technical Reports Server (NTRS)
Cleaveland, Rance; Luettgen, Gerald; Natarajan, V.
1999-01-01
This paper surveys the semantic ramifications of extending traditional process algebras with notions of priority that allow for some transitions to be given precedence over others. These enriched formalisms allow one to model system features such as interrupts, prioritized choice, or real-time behavior. Approaches to priority in process algebras can be classified according to whether the induced notion of preemption on transitions is global or local and whether priorities are static or dynamic. Early work in the area concentrated on global pre-emption and static priorities and led to formalisms for modeling interrupts and aspects of real-time, such as maximal progress, in centralized computing environments. More recent research has investigated localized notions of pre-emption in which the distribution of systems is taken into account, as well as dynamic priority approaches, i.e., those where priority values may change as systems evolve. The latter allows one to model behavioral phenomena such as scheduling algorithms and also enables the efficient encoding of real-time semantics. Technically, this paper studies the different models of priorities by presenting extensions of Milner's Calculus of Communicating Systems (CCS) with static and dynamic priority as well as with notions of global and local pre- emption. In each case the operational semantics of CCS is modified appropriately, behavioral theories based on strong and weak bisimulation are given, and related approaches for different process-algebraic settings are discussed.
On the cohomology of Leibniz conformal algebras
NASA Astrophysics Data System (ADS)
Zhang, Jiao
2015-04-01
We construct a new cohomology complex of Leibniz conformal algebras with coefficients in a representation instead of a module. The low-dimensional cohomology groups of this complex are computed. Meanwhile, we construct a Leibniz algebra from a Leibniz conformal algebra and prove that the category of Leibniz conformal algebras is equivalent to the category of equivalence classes of formal distribution Leibniz algebras.
Graphs and Matroids Weighted in a Bounded Incline Algebra
Lu, Ling-Xia; Zhang, Bei
2014-01-01
Firstly, for a graph weighted in a bounded incline algebra (or called a dioid), a longest path problem (LPP, for short) is presented, which can be considered the uniform approach to the famous shortest path problem, the widest path problem, and the most reliable path problem. The solutions for LPP and related algorithms are given. Secondly, for a matroid weighted in a linear matroid, the maximum independent set problem is studied. PMID:25126607
Graphs and matroids weighted in a bounded incline algebra.
Lu, Ling-Xia; Zhang, Bei
2014-01-01
Firstly, for a graph weighted in a bounded incline algebra (or called a dioid), a longest path problem (LPP, for short) is presented, which can be considered the uniform approach to the famous shortest path problem, the widest path problem, and the most reliable path problem. The solutions for LPP and related algorithms are given. Secondly, for a matroid weighted in a linear matroid, the maximum independent set problem is studied. PMID:25126607
Graphs and matroids weighted in a bounded incline algebra.
Lu, Ling-Xia; Zhang, Bei
2014-01-01
Firstly, for a graph weighted in a bounded incline algebra (or called a dioid), a longest path problem (LPP, for short) is presented, which can be considered the uniform approach to the famous shortest path problem, the widest path problem, and the most reliable path problem. The solutions for LPP and related algorithms are given. Secondly, for a matroid weighted in a linear matroid, the maximum independent set problem is studied.
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s. PMID:26806075
Verburgt, Lukas M
2016-01-01
This paper provides a detailed account of the period of the complex history of British algebra and geometry between the publication of George Peacock's Treatise on Algebra in 1830 and William Rowan Hamilton's paper on quaternions of 1843. During these years, Duncan Farquharson Gregory and William Walton published several contributions on 'algebraical geometry' and 'geometrical algebra' in the Cambridge Mathematical Journal. These contributions enabled them not only to generalize Peacock's symbolical algebra on the basis of geometrical considerations, but also to initiate the attempts to question the status of Euclidean space as the arbiter of valid geometrical interpretations. At the same time, Gregory and Walton were bound by the limits of symbolical algebra that they themselves made explicit; their work was not and could not be the 'abstract algebra' and 'abstract geometry' of figures such as Hamilton and Cayley. The central argument of the paper is that an understanding of the contributions to 'algebraical geometry' and 'geometrical algebra' of the second generation of 'scientific' symbolical algebraists is essential for a satisfactory explanation of the radical transition from symbolical to abstract algebra that took place in British mathematics in the 1830s-1840s.
Second-Order Algebraic Theories
NASA Astrophysics Data System (ADS)
Fiore, Marcelo; Mahmoud, Ola
Fiore and Hur [10] recently introduced a conservative extension of universal algebra and equational logic from first to second order. Second-order universal algebra and second-order equational logic respectively provide a model theory and a formal deductive system for languages with variable binding and parameterised metavariables. This work completes the foundations of the subject from the viewpoint of categorical algebra. Specifically, the paper introduces the notion of second-order algebraic theory and develops its basic theory. Two categorical equivalences are established: at the syntactic level, that of second-order equational presentations and second-order algebraic theories; at the semantic level, that of second-order algebras and second-order functorial models. Our development includes a mathematical definition of syntactic translation between second-order equational presentations. This gives the first formalisation of notions such as encodings and transforms in the context of languages with variable binding.
Native and Nonnative Use of Multi-Word vs. One-Word Verbs
ERIC Educational Resources Information Center
Siyanova, Anna; Schmitt, Norbert
2007-01-01
One of the choices available in English is between one-word verbs (train at the gym) and their multi-word counterparts (work out at the gym). Multi-word verbs tend to be colloquial in tone and are a particular feature of informal spoken discourse. Previous research suggests that English learners often have problems with multi-word verbs, and may…
ERIC Educational Resources Information Center
Novotna, Jarmila; Hoch, Maureen
2008-01-01
Many students have difficulties with basic algebraic concepts at high school and at university. In this paper two levels of algebraic structure sense are defined: for high school algebra and for university algebra. We suggest that high school algebra structure sense components are sub-components of some university algebra structure sense…
2-Local derivations on matrix algebras over semi-prime Banach algebras and on AW*-algebras
NASA Astrophysics Data System (ADS)
Ayupov, Shavkat; Kudaybergenov, Karimbergen
2016-03-01
The paper is devoted to 2-local derivations on matrix algebras over unital semi-prime Banach algebras. For a unital semi-prime Banach algebra A with the inner derivation property we prove that any 2-local derivation on the algebra M 2n (A), n ≥ 2, is a derivation. We apply this result to AW*-algebras and show that any 2-local derivation on an arbitrary AW*-algebra is a derivation.
Algebraic Modeling of Information Retrieval in XML Documents
NASA Astrophysics Data System (ADS)
Georgiev, Bozhidar; Georgieva, Adriana
2009-11-01
This paper presents an information retrieval approach in XML documents using tools, based on the linear algebra. The well-known transformation languages as XSLT (XPath) are grounded on the features of higher-order logic for manipulating hierarchical trees. The presented conception is compared to existing higher-order logic formalisms, where the queries are realized by both languages XSLT and XPath. The possibilities of the proposed linear algebraic model combined with hierarchy data models permit more efficient solutions for searching, extracting and manipulating semi-structured data with hierarchical structures avoiding the global navigation over the XML tree components. The main purpose of this algebraic model representation, applied to the hierarchical relationships in the XML data structures, is to make the implementation of linear algebra tools possible for XML data manipulations and to eliminate existing problems, related to regular grammars theory and also to avoid the difficulties, connected with higher -order logic (first-order logic, monadic second- order logic etc.).
Plethystic algebras and vector symmetric functions.
Rota, G C; Stein, J A
1994-01-01
An isomorphism is established between the plethystic Hopf algebra Pleth(Super[L]) and the algebra of vector symmetric functions. The Hall inner product of symmetric function theory is extended to the Hopf algebra Pleth(Super[L]). PMID:11607504
Algebra and Algebraic Thinking in School Math: 70th YB
ERIC Educational Resources Information Center
National Council of Teachers of Mathematics, 2008
2008-01-01
Algebra is no longer just for college-bound students. After a widespread push by the National Council of Teachers of Mathematics (NCTM) and teachers across the country, algebra is now a required part of most curricula. However, students' standardized test scores are not at the level they should be. NCTM's seventieth yearbook takes a look at the…
Abstract Algebra to Secondary School Algebra: Building Bridges
ERIC Educational Resources Information Center
Christy, Donna; Sparks, Rebecca
2015-01-01
The authors have experience with secondary mathematics teacher candidates struggling to make connections between the theoretical abstract algebra course they take as college students and the algebra they will be teaching in secondary schools. As a mathematician and a mathematics educator, the authors collaborated to create and implement a…
Handheld Computer Algebra Systems in the Pre-Algebra Classroom
ERIC Educational Resources Information Center
Gantz, Linda Ann Galofaro
2010-01-01
This mixed method analysis sought to investigate several aspects of student learning in pre-algebra through the use of computer algebra systems (CAS) as opposed to non-CAS learning. This research was broken into two main parts, one which compared results from both the experimental group (instruction using CAS, N = 18) and the control group…
Hidden algebra method (quasi-exact-solvability in quantum mechanics)
Turbiner, Alexander
1996-02-20
A general introduction to quasi-exactly-solvable problems of quantum mechanics is presented. Main attention is given to multidimensional quasi-exactly-solvable and exactly-solvable Schroedinger operators. Exact-solvability of the Calogero and Sutherland N-body problems ass ociated with an existence of the hidden algebra slN is discussed extensively.
Clearing the Fog from the Undergraduate Course in Linear Algebra
ERIC Educational Resources Information Center
Scott, Damon
2007-01-01
For over a decade it has been a common observation that a "fog" passes over the course in linear algebra once abstract vector spaces are presented. See [2, 3]. We show how this fog may be cleared by having the students translate "abstract" vector-space problems to isomorphic "concrete" settings, solve the "concrete" problem either by hand or with…
Statecharts Via Process Algebra
NASA Technical Reports Server (NTRS)
Luttgen, Gerald; vonderBeeck, Michael; Cleaveland, Rance
1999-01-01
Statecharts is a visual language for specifying the behavior of reactive systems. The Language extends finite-state machines with concepts of hierarchy, concurrency, and priority. Despite its popularity as a design notation for embedded system, precisely defining its semantics has proved extremely challenging. In this paper, a simple process algebra, called Statecharts Process Language (SPL), is presented, which is expressive enough for encoding Statecharts in a structure-preserving and semantic preserving manner. It is establish that the behavioral relation bisimulation, when applied to SPL, preserves Statecharts semantics
Linear Algebra and Image Processing
ERIC Educational Resources Information Center
Allali, Mohamed
2010-01-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty. (Contains 2 tables and 11 figures.)
Linear algebra and image processing
NASA Astrophysics Data System (ADS)
Allali, Mohamed
2010-09-01
We use the computing technology digital image processing (DIP) to enhance the teaching of linear algebra so as to make the course more visual and interesting. Certainly, this visual approach by using technology to link linear algebra to DIP is interesting and unexpected to both students as well as many faculty.
A Programmed Course in Algebra.
ERIC Educational Resources Information Center
Mewborn, Ancel C.; Hively, Wells II
This programed textbook consists of short sections of text interspersed with questions designed to aid the student in understanding the material. The course is designed to increase the student's understanding of some of the basic ideas of algebra. Some general experience and manipulative skill with respect to high school algebra is assumed.…
ERIC Educational Resources Information Center
1997
Astro Algebra is one of six titles in the Mighty Math Series from Edmark, a comprehensive line of math software for students from kindergarten through ninth grade. Many of the activities in Astro Algebra contain a unique technology that uses the computer to help students make the connection between concrete and abstract mathematics. This software…
Gamow functionals on operator algebras
NASA Astrophysics Data System (ADS)
Castagnino, M.; Gadella, M.; Betán, R. Id; Laura, R.
2001-11-01
We obtain the precise form of two Gamow functionals representing the exponentially decaying part of a quantum resonance and its mirror image that grows exponentially, as a linear, positive and continuous functional on an algebra containing observables. These functionals do not admit normalization and, with an appropriate choice of the algebra, are time reversal of each other.
Online Algebraic Tools for Teaching
ERIC Educational Resources Information Center
Kurz, Terri L.
2011-01-01
Many free online tools exist to complement algebraic instruction at the middle school level. This article presents findings that analyzed the features of algebraic tools to support learning. The findings can help teachers select appropriate tools to facilitate specific topics. (Contains 1 table and 4 figures.)
ERIC Educational Resources Information Center
Instructional Objectives Exchange, Los Angeles, CA.
A complete set of behavioral objectives for first-year algebra taught in any of grades 8 through 12 is presented. Three to six sample test items and answers are provided for each objective. Objectives were determined by surveying the most used secondary school algebra textbooks. Fourteen major categories are included: (1) whole numbers--operations…
Elementary maps on nest algebras
NASA Astrophysics Data System (ADS)
Li, Pengtong
2006-08-01
Let , be algebras and let , be maps. An elementary map of is an ordered pair (M,M*) such that for all , . In this paper, the general form of surjective elementary maps on standard subalgebras of nest algebras is described. In particular, such maps are automatically additive.
Condensing Algebra for Technical Mathematics.
ERIC Educational Resources Information Center
Greenfield, Donald R.
Twenty Algebra-Packets (A-PAKS) were developed by the investigator for technical education students at the community college level. Each packet contained a statement of rationale, learning objectives, performance activities, performance test, and performance test answer key. The A-PAKS condensed the usual sixteen weeks of algebra into a six-week…
ERIC Educational Resources Information Center
Buerman, Margaret
2007-01-01
Finding real-world examples for middle school algebra classes can be difficult but not impossible. As we strive to accomplish teaching our students how to solve and graph equations, we neglect to teach the big ideas of algebra. One of those big ideas is functions. This article gives three examples of functions that are found in Arches National…
What Homophones Say about Words
Dautriche, Isabelle; Chemla, Emmanuel
2016-01-01
The number of potential meanings for a new word is astronomic. To make the word-learning problem tractable, one must restrict the hypothesis space. To do so, current word learning accounts often incorporate constraints about cognition or about the mature lexicon directly in the learning device. We are concerned with the convexity constraint, which holds that concepts (privileged sets of entities that we think of as “coherent”) do not have gaps (if A and B belong to a concept, so does any entity “between” A and B). To leverage from it a linguistic constraint, learning algorithms have percolated this constraint from concepts, to word forms: some algorithms rely on the possibility that word forms are associated with convex sets of objects. Yet this does have to be the case: homophones are word forms associated with two separate words and meanings. Two sets of experiments show that when evidence suggests that a novel label is associated with a disjoint (non-convex) set of objects, either a) because there is a gap in conceptual space between the learning exemplars for a given word or b) because of the intervention of other lexical items in that gap, adults prefer to postulate homophony, where a single word form is associated with two separate words and meanings, rather than inferring that the word could have a disjunctive, discontinuous meaning. These results about homophony must be integrated to current word learning algorithms. We conclude by arguing for a weaker specialization of word learning algorithms, which too often could miss important constraints by focusing on a restricted empirical basis (e.g., non-homophonous content words). PMID:27583384
What Homophones Say about Words.
Dautriche, Isabelle; Chemla, Emmanuel
2016-01-01
The number of potential meanings for a new word is astronomic. To make the word-learning problem tractable, one must restrict the hypothesis space. To do so, current word learning accounts often incorporate constraints about cognition or about the mature lexicon directly in the learning device. We are concerned with the convexity constraint, which holds that concepts (privileged sets of entities that we think of as "coherent") do not have gaps (if A and B belong to a concept, so does any entity "between" A and B). To leverage from it a linguistic constraint, learning algorithms have percolated this constraint from concepts, to word forms: some algorithms rely on the possibility that word forms are associated with convex sets of objects. Yet this does have to be the case: homophones are word forms associated with two separate words and meanings. Two sets of experiments show that when evidence suggests that a novel label is associated with a disjoint (non-convex) set of objects, either a) because there is a gap in conceptual space between the learning exemplars for a given word or b) because of the intervention of other lexical items in that gap, adults prefer to postulate homophony, where a single word form is associated with two separate words and meanings, rather than inferring that the word could have a disjunctive, discontinuous meaning. These results about homophony must be integrated to current word learning algorithms. We conclude by arguing for a weaker specialization of word learning algorithms, which too often could miss important constraints by focusing on a restricted empirical basis (e.g., non-homophonous content words). PMID:27583384
Thermodynamics. [algebraic structure
NASA Technical Reports Server (NTRS)
Zeleznik, F. J.
1976-01-01
The fundamental structure of thermodynamics is purely algebraic, in the sense of atopological, and it is also independent of partitions, composite systems, the zeroth law, and entropy. The algebraic structure requires the notion of heat, but not the first law. It contains a precise definition of entropy and identifies it as a purely mathematical concept. It also permits the construction of an entropy function from heat measurements alone when appropriate conditions are satisfied. Topology is required only for a discussion of the continuity of thermodynamic properties, and then the weak topology is the relevant topology. The integrability of the differential form of the first law can be examined independently of Caratheodory's theorem and his inaccessibility axiom. Criteria are established by which one can determine when an integrating factor can be made intensive and the pseudopotential extensive and also an entropy. Finally, a realization of the first law is constructed which is suitable for all systems whether they are solids or fluids, whether they do or do not exhibit chemical reactions, and whether electromagnetic fields are or are not present.
ERIC Educational Resources Information Center
Education Commission of the States, Denver, CO.
This paper provides an overview of Fast ForWord, a CD-ROM and Internet-based training program for children (pre-K to grade 8) with language and reading problems that helps children rapidly build oral language comprehension and other critical skills necessary for learning to read or becoming a better reader. With the help of computers, speech…
Word Processing Curriculum Guide.
ERIC Educational Resources Information Center
Anderson, Marcia A.; Kusek, Robert W.
A combination of facts, examples, models, tools, and sources useful in developing and teaching word processing (WP) programs is provided in this guide. Eight sections are included. Sections 1 and 2 present introductory information on WP (e.g., history, five phases of WP, problems occurring in WP offices, factors of people, procedures, and…
The coquaternion algebra and complex partial differential equations
NASA Astrophysics Data System (ADS)
Dimiev, Stancho; Konstantinov, Mihail; Todorov, Vladimir
2009-11-01
In this paper we consider the problem of differentiation of coquaternionic functions. Let us recall that coquaternions are elements of an associative non-commutative real algebra with zero divisor, introduced by James Cockle (1849) under the name of split-quaternions or coquaternions. Developing two type complex representations for Cockle algebra (complex and paracomplex ones) we present the problem in a non-commutative form of the δ¯-type holomorphy. We prove that corresponding differentiable coquaternionic functions, smooth and analytic, satisfy PDE of complex, and respectively of real variables. Applications for coquaternionic polynomials are sketched.
Element Agglomeration Algebraic Multilevel Monte-Carlo Library
2015-02-19
ElagMC is a parallel C++ library for Multilevel Monte Carlo simulations with algebraically constructed coarse spaces. ElagMC enables Multilevel variance reduction techniques in the context of general unstructured meshes by using the specialized element-based agglomeration techniques implemented in ELAG (the Element-Agglomeration Algebraic Multigrid and Upscaling Library developed by U. Villa and P. Vassilevski and currently under review for public release). The ElabMC library can support different type of deterministic problems, including mixed finite element discretizations of subsurface flow problems.
Element Agglomeration Algebraic Multilevel Monte-Carlo Library
2015-02-19
ElagMC is a parallel C++ library for Multilevel Monte Carlo simulations with algebraically constructed coarse spaces. ElagMC enables Multilevel variance reduction techniques in the context of general unstructured meshes by using the specialized element-based agglomeration techniques implemented in ELAG (the Element-Agglomeration Algebraic Multigrid and Upscaling Library developed by U. Villa and P. Vassilevski and currently under review for public release). The ElabMC library can support different type of deterministic problems, including mixed finite element discretizationsmore » of subsurface flow problems.« less
Computer algebra methods in the study of nonlinear differential systems
NASA Astrophysics Data System (ADS)
Irtegov, V. D.; Titorenko, T. N.
2013-06-01
Some issues concerning computer algebra methods as applied to the qualitative analysis of differential equations with first integrals are discussed. The problems of finding stationary sets and analyzing their stability and bifurcations are considered. Special attention is given to algorithms for finding and analyzing peculiar stationary sets. It is shown that computer algebra tools, combined with qualitative analysis methods for differential equations, make it possible not only to enhance the computational efficiency of classical algorithms, but also to implement new approaches to the solution of well-known problems and, in this way, to obtain new results.
The noncommutative Poisson bracket and the deformation of the family algebras
Wei, Zhaoting
2015-07-15
The family algebras are introduced by Kirillov in 2000. In this paper, we study the noncommutative Poisson bracket P on the classical family algebra C{sub τ}(g). We show that P controls the first-order 1-parameter formal deformation from C{sub τ}(g) to Q{sub τ}(g) where the latter is the quantum family algebra. Moreover, we will prove that the noncommutative Poisson bracket is in fact a Hochschild 2-coboundary, and therefore, the deformation is infinitesimally trivial. In the last part of this paper, we discuss the relation between Mackey’s analogue and the quantization problem of the family algebras.
Sixth SIAM conference on applied linear algebra: Final program and abstracts. Final technical report
1997-12-31
Linear algebra plays a central role in mathematics and applications. The analysis and solution of problems from an amazingly wide variety of disciplines depend on the theory and computational techniques of linear algebra. In turn, the diversity of disciplines depending on linear algebra also serves to focus and shape its development. Some problems have special properties (numerical, structural) that can be exploited. Some are simply so large that conventional approaches are impractical. New computer architectures motivate new algorithms, and fresh ways to look at old ones. The pervasive nature of linear algebra in analyzing and solving problems means that people from a wide spectrum--universities, industrial and government laboratories, financial institutions, and many others--share an interest in current developments in linear algebra. This conference aims to bring them together for their mutual benefit. Abstracts of papers presented are included.
SD-CAS: Spin Dynamics by Computer Algebra System.
Filip, Xenia; Filip, Claudiu
2010-11-01
A computer algebra tool for describing the Liouville-space quantum evolution of nuclear 1/2-spins is introduced and implemented within a computational framework named Spin Dynamics by Computer Algebra System (SD-CAS). A distinctive feature compared with numerical and previous computer algebra approaches to solving spin dynamics problems results from the fact that no matrix representation for spin operators is used in SD-CAS, which determines a full symbolic character to the performed computations. Spin correlations are stored in SD-CAS as four-entry nested lists of which size increases linearly with the number of spins into the system and are easily mapped into analytical expressions in terms of spin operator products. For the so defined SD-CAS spin correlations a set of specialized functions and procedures is introduced that are essential for implementing basic spin algebra operations, such as the spin operator products, commutators, and scalar products. They provide results in an abstract algebraic form: specific procedures to quantitatively evaluate such symbolic expressions with respect to the involved spin interaction parameters and experimental conditions are also discussed. Although the main focus in the present work is on laying the foundation for spin dynamics symbolic computation in NMR based on a non-matrix formalism, practical aspects are also considered throughout the theoretical development process. In particular, specific SD-CAS routines have been implemented using the YACAS computer algebra package (http://yacas.sourceforge.net), and their functionality was demonstrated on a few illustrative examples.
SD-CAS: Spin Dynamics by Computer Algebra System
NASA Astrophysics Data System (ADS)
Filip, Xenia; Filip, Claudiu
2010-11-01
A computer algebra tool for describing the Liouville-space quantum evolution of nuclear 1/2-spins is introduced and implemented within a computational framework named Spin Dynamics by Computer Algebra System (SD-CAS). A distinctive feature compared with numerical and previous computer algebra approaches to solving spin dynamics problems results from the fact that no matrix representation for spin operators is used in SD-CAS, which determines a full symbolic character to the performed computations. Spin correlations are stored in SD-CAS as four-entry nested lists of which size increases linearly with the number of spins into the system and are easily mapped into analytical expressions in terms of spin operator products. For the so defined SD-CAS spin correlations a set of specialized functions and procedures is introduced that are essential for implementing basic spin algebra operations, such as the spin operator products, commutators, and scalar products. They provide results in an abstract algebraic form: specific procedures to quantitatively evaluate such symbolic expressions with respect to the involved spin interaction parameters and experimental conditions are also discussed. Although the main focus in the present work is on laying the foundation for spin dynamics symbolic computation in NMR based on a non-matrix formalism, practical aspects are also considered throughout the theoretical development process. In particular, specific SD-CAS routines have been implemented using the YACAS computer algebra package (http://yacas.sourceforge.net), and their functionality was demonstrated on a few illustrative examples.
Using computer algebra and SMT solvers in algebraic biology
NASA Astrophysics Data System (ADS)
Pineda Osorio, Mateo
2014-05-01
Biologic processes are represented as Boolean networks, in a discrete time. The dynamics within these networks are approached with the help of SMT Solvers and the use of computer algebra. Software such as Maple and Z3 was used in this case. The number of stationary states for each network was calculated. The network studied here corresponds to the immune system under the effects of drastic mood changes. Mood is considered as a Boolean variable that affects the entire dynamics of the immune system, changing the Boolean satisfiability and the number of stationary states of the immune network. Results obtained show Z3's great potential as a SMT Solver. Some of these results were verified in Maple, even though it showed not to be as suitable for the problem approach. The solving code was constructed using Z3-Python and Z3-SMT-LiB. Results obtained are important in biology systems and are expected to help in the design of immune therapies. As a future line of research, more complex Boolean network representations of the immune system as well as the whole psychological apparatus are suggested.
ERIC Educational Resources Information Center
Gonzalez-Vega, Laureano
1999-01-01
Using a Computer Algebra System (CAS) to help with the teaching of an elementary course in linear algebra can be one way to introduce computer algebra, numerical analysis, data structures, and algorithms. Highlights the advantages and disadvantages of this approach to the teaching of linear algebra. (Author/MM)
Student Learning in Linear Algebra: The Gateways To Advance Mathematical Thinking Project.
ERIC Educational Resources Information Center
Manes, Michelle
This document provides a preliminary report of the study Gateways To Advance Mathematical Thinking (GAMT) run by Educational Development Center, Inc. (EDC). The study was designed to see what types of reasoning students who have recently completed a linear algebra course apply to problems in algebraic thinking. Student interviews were used as the…
Activities for Students: Biology as a Source for Algebra Equations--The Heart
ERIC Educational Resources Information Center
Horak, Virginia M.
2005-01-01
The high school course that integrated first year algebra with an introductory environmental biology/anatomy and physiology course, in order to solve algebra problems is discussed. Lessons and activities for the course were taken by identifying the areas where mathematics and biology content intervenes may help students understand biology concepts…
A Computer Algebra Approach to Solving Chemical Equilibria in General Chemistry
ERIC Educational Resources Information Center
Kalainoff, Melinda; Lachance, Russ; Riegner, Dawn; Biaglow, Andrew
2012-01-01
In this article, we report on a semester-long study of the incorporation into our general chemistry course, of advanced algebraic and computer algebra techniques for solving chemical equilibrium problems. The method presented here is an alternative to the commonly used concentration table method for describing chemical equilibria in general…
An Example of Competence-Based Learning: Use of Maxima in Linear Algebra for Engineers
ERIC Educational Resources Information Center
Diaz, Ana; Garcia, Alfonsa; de la Villa, Agustin
2011-01-01
This paper analyses the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is…
Constraints on the Meanings of Words.
ERIC Educational Resources Information Center
Soja, N.; And Others
Between their second and fifth years, young children learn approximately 15 new words a day. For every word the child hears, he or she must choose the correct referent out of an infinite set of candidates. An important problem for developmental psychologists is to understand the principles that limit the child's hypotheses about word meanings. A…
ERIC Educational Resources Information Center
Parmer, Lavada Jacumin; Thames, Dana G.; Kazelskis, Richard
A study examined the effectiveness of an integrated language arts instructional format for teaching reading compared with the effectiveness of the typical traditional reading program. The study investigated the effectiveness of approaches that are representative of both viewpoints of the reading process (i.e., word recognition and the construction…
Quantum algebra of N superspace
Hatcher, Nicolas; Restuccia, A.; Stephany, J.
2007-08-15
We identify the quantum algebra of position and momentum operators for a quantum system bearing an irreducible representation of the super Poincare algebra in the N>1 and D=4 superspace, both in the case where there are no central charges in the algebra, and when they are present. This algebra is noncommutative for the position operators. We use the properties of superprojectors acting on the superfields to construct explicit position and momentum operators satisfying the algebra. They act on the projected wave functions associated to the various supermultiplets with defined superspin present in the representation. We show that the quantum algebra associated to the massive superparticle appears in our construction and is described by a supermultiplet of superspin 0. This result generalizes the construction for D=4, N=1 reported recently. For the case N=2 with central charges, we present the equivalent results when the central charge and the mass are different. For the {kappa}-symmetric case when these quantities are equal, we discuss the reduction to the physical degrees of freedom of the corresponding superparticle and the construction of the associated quantum algebra.
Constraint algebra in bigravity
Soloviev, V. O.
2015-07-15
The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.
Constraint algebra in bigravity
NASA Astrophysics Data System (ADS)
Soloviev, V. O.
2015-07-01
The number of degrees of freedom in bigravity theory is found for a potential of general form and also for the potential proposed by de Rham, Gabadadze, and Tolley (dRGT). This aim is pursued via constructing a Hamiltonian formalismand studying the Poisson algebra of constraints. A general potential leads to a theory featuring four first-class constraints generated by general covariance. The vanishing of the respective Hessian is a crucial property of the dRGT potential, and this leads to the appearance of two additional second-class constraints and, hence, to the exclusion of a superfluous degree of freedom—that is, the Boulware—Deser ghost. The use of a method that permits avoiding an explicit expression for the dRGT potential is a distinctive feature of the present study.
Algebraic approximations for transcendental equations with applications in nanophysics
NASA Astrophysics Data System (ADS)
Barsan, Victor
2015-09-01
Using algebraic approximations of trigonometric or hyperbolic functions, a class of transcendental equations can be transformed in tractable, algebraic equations. Studying transcendental equations this way gives the eigenvalues of Sturm-Liouville problems associated to wave equation, mainly to Schroedinger equation; these algebraic approximations provide approximate analytical expressions for the energy of electrons and phonons in quantum wells, quantum dots (QDs) and quantum wires, in the frame of one-particle models of such systems. The advantage of this approach, compared to the numerical calculations, is that the final result preserves the functional dependence on the physical parameters of the problem. The errors of this method, situated between some few percentages and ?, are carefully analysed. Several applications, for quantum wells, QDs and quantum wires, are presented.
Readiness and Preparation for Beginning Algebra.
ERIC Educational Resources Information Center
Rotman, Jack W.
Drawing from experience at Lansing Community College (LCC), this paper discusses how to best prepare students for success in a beginning algebra course. First, an overview is presented of LCC's developmental math sequence, which includes Basic Arithmetic (MTH 008), Pre-Algebra (MTH 009), Beginning Algebra (MTH 012), and Intermediate Algebra (MTH…
Hopf algebras and Dyson-Schwinger equations
NASA Astrophysics Data System (ADS)
Weinzierl, Stefan
2016-06-01
In this paper I discuss Hopf algebras and Dyson-Schwinger equations. This paper starts with an introduction to Hopf algebras, followed by a review of the contribution and application of Hopf algebras to particle physics. The final part of the paper is devoted to the relation between Hopf algebras and Dyson-Schwinger equations.
Two-parameter twisted quantum affine algebras
NASA Astrophysics Data System (ADS)
Jing, Naihuan; Zhang, Honglian
2016-09-01
We establish Drinfeld realization for the two-parameter twisted quantum affine algebras using a new method. The Hopf algebra structure for Drinfeld generators is given for both untwisted and twisted two-parameter quantum affine algebras, which include the quantum affine algebras as special cases.
ERIC Educational Resources Information Center
Sullivan, Patrick
2013-01-01
The purpose of this study is to examine the nature of what students notice about symbols and use as they solve unfamiliar algebra problems based on familiar algebra concepts and involving symbolic inscriptions. The researcher conducted a study of students at three levels of algebra exposure: (a) students enrolled in a high school pre-calculus…
ERIC Educational Resources Information Center
Khajarian, Seta
2011-01-01
Algebra is a branch in mathematics and taking Algebra in middle school is often a gateway to advanced courses in high school. The problem is that the United States and Lebanon had low scores in Algebra in the 2007 Trends in Mathematics and Sciences Study (TIMSS), an international assessment administered to 4th and 8th graders every 4 years. On the…
ERIC Educational Resources Information Center
Egodawatte, Gunawardena; Stoilescu, Dorian
2015-01-01
The purpose of this mixed-method study was to investigate grade 11 university/college stream mathematics students' difficulties in applying conceptual knowledge, procedural skills, strategic competence, and algebraic thinking in solving routine (instructional) algebraic problems. A standardized algebra test was administered to thirty randomly…
Compatible Relaxation and Coarsening in Algebraic Multigrid
Brannick, J J; Falgout, R D
2009-09-22
We introduce a coarsening algorithm for algebraic multigrid (AMG) based on the concept of compatible relaxation (CR). The algorithm is significantly different from standard methods, most notably because it does not rely on any notion of strength of connection. We study its behavior on a number of model problems, and evaluate the performance of an AMG algorithm that incorporates the coarsening approach. Lastly, we introduce a variant of CR that provides a sharper metric of coarse-grid quality and demonstrate its potential with two simple examples.
Cartooning in Algebra and Calculus
ERIC Educational Resources Information Center
Moseley, L. Jeneva
2014-01-01
This article discusses how teachers can create cartoons for undergraduate math classes, such as college algebra and basic calculus. The practice of cartooning for teaching can be helpful for communication with students and for students' conceptual understanding.
NASA Technical Reports Server (NTRS)
Klumpp, A. R.; Lawson, C. L.
1988-01-01
Routines provided for common scalar, vector, matrix, and quaternion operations. Computer program extends Ada programming language to include linear-algebra capabilities similar to HAS/S programming language. Designed for such avionics applications as software for Space Station.
GCD, LCM, and Boolean Algebra?
ERIC Educational Resources Information Center
Cohen, Martin P.; Juraschek, William A.
1976-01-01
This article investigates the algebraic structure formed when the process of finding the greatest common divisor and the least common multiple are considered as binary operations on selected subsets of positive integers. (DT)
Leikin, Mark; Waisman, Ilana; Shaul, Shelley; Leikin, Roza
2014-03-01
This paper presents a small part of a larger interdisciplinary study that investigates brain activity (using event related potential methodology) of male adolescents when solving mathematical problems of different types. The study design links mathematics education research with neurocognitive studies. In this paper we performed a comparative analysis of brain activity associated with the translation from visual to symbolic representations of mathematical objects in algebra and geometry. Algebraic tasks require translation from graphical to symbolic representation of a function, whereas tasks in geometry require translation from a drawing of a geometric figure to a symbolic representation of its property. The findings demonstrate that electrical activity associated with the performance of geometrical tasks is stronger than that associated with solving algebraic tasks. Additionally, we found different scalp topography of the brain activity associated with algebraic and geometric tasks. Based on these results, we argue that problem solving in algebra and geometry is associated with different patterns of brain activity.
Rumelhart, D.E.; Skokowski, P.G.; Martin, B.O.
1995-05-01
In this project we have developed a language model based on Artificial Neural Networks (ANNs) for use in conjunction with automatic textual search or speech recognition systems. The model can be trained on large corpora of text to produce probability estimates that would improve the ability of systems to identify words in a sentence given partial contextual information. The model uses a gradient-descent learning procedure to develop a metric of similarity among terms in a corpus, based on context. Using lexical categories based on this metric, a network can then be trained to do serial word probability estimation. Such a metric can also be used to improve the performance of topic-based search by allowing retrieval of information that is related to desired topics even if no obvious set of key words unites all the retrieved items.
Hopf algebras and topological recursion
NASA Astrophysics Data System (ADS)
Esteves, João N.
2015-11-01
We consider a model for topological recursion based on the Hopf algebra of planar binary trees defined by Loday and Ronco (1998 Adv. Math. 139 293-309 We show that extending this Hopf algebra by identifying pairs of nearest neighbor leaves, and thus producing graphs with loops, we obtain the full recursion formula discovered by Eynard and Orantin (2007 Commun. Number Theory Phys. 1 347-452).
Sjaardema, G.; Gilkey, A.; Smith, M.; Forsythe, C.
2005-04-11
The ALGEBRA program allows the user to manipulate data from a finite element analysis before it is plotted. The finite element output data is in the form of variable values (e.g., stress, strain, and velocity components) in an EXODUS II database. The ALGEBRA program evaluates user-supplied functions of the data and writes the results to an output EXODUS II database that can be read by plot programs.
Algebraic Systems and Pushdown Automata
NASA Astrophysics Data System (ADS)
Petre, Ion; Salomaa, Arto
We concentrate in this chapter on the core aspects of algebraic series, pushdown automata, and their relation to formal languages. We choose to follow here a presentation of their theory based on the concept of properness. We introduce in Sect. 2 some auxiliary notions and results needed throughout the chapter, in particular the notions of discrete convergence in semirings and C-cycle free infinite matrices. In Sect. 3 we introduce the algebraic power series in terms of algebraic systems of equations. We focus on interconnections with context-free grammars and on normal forms. We then conclude the section with a presentation of the theorems of Shamir and Chomsky-Schützenberger. We discuss in Sect. 4 the algebraic and the regulated rational transductions, as well as some representation results related to them. Section 5 is dedicated to pushdown automata and focuses on the interconnections with classical (non-weighted) pushdown automata and on the interconnections with algebraic systems. We then conclude the chapter with a brief discussion of some of the other topics related to algebraic systems and pushdown automata.
Mathematics as Problem Solving.
ERIC Educational Resources Information Center
Soifer, Alexander
This book contains about 200 problems. It is suggested that it be used by students, teachers or anyone interested in exploring mathematics. In addition to a general discussion on problem solving, there are problems concerned with number theory, algebra, geometry, and combinatorics. (PK)
Algebraic formulation of quantum theory, particle identity and entanglement
NASA Astrophysics Data System (ADS)
Govindarajan, T. R.
2016-08-01
Quantum theory as formulated in conventional framework using statevectors in Hilbert spaces misses the statistical nature of the underlying quantum physics. Formulation using operators 𝒞∗ algebra and density matrices appropriately captures this feature in addition leading to the correct formulation of particle identity. In this framework, Hilbert space is an emergent concept. Problems related to anomalies and quantum epistemology are discussed.
Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School.
ERIC Educational Resources Information Center
Carpenter, Thomas P.; Franke, Megan Loef; Levi, Linda
This book is designed to help teachers understand children's intuitive problem solving and computational processes and to figure out how to use that knowledge to enhance students' understanding of arithmetic. This book provides numerous examples of classroom dialogues that indicate how algebraic ideas emerge in children's thinking and what…
Undergraduate Mathematics Students' Emotional Experiences in Linear Algebra Courses
ERIC Educational Resources Information Center
Martínez-Sierra, Gustavo; García-González, María del Socorro
2016-01-01
Little is known about students' emotions in the field of Mathematics Education that go beyond students' emotions in problem solving. To start filling this gap this qualitative research has the aim to identify emotional experiences of undergraduate mathematics students in Linear Algebra courses. In order to obtain data, retrospective focus group…
Fixing Ganache: Another Real-Life Use for Algebra
ERIC Educational Resources Information Center
Kalman, Adam M.
2011-01-01
This article presents a real-world application of proportional reasoning and equation solving. The author describes how students adjust ingredient amounts in a recipe for chocolate ganache. Using this real-world scenario provided students an opportunity to solve a difficult and nonstandard algebra problem, a lot of practice with fractions, a…
How to write fast and clear parallel programs using algebra
Stiller, L. Johns Hopkins Univ., Baltimore, MD )
1992-01-01
An algebraic method for the design of efficient and easy to port codes for parallel machines is described. The method was applied to speed up and to clarify certain communication functions, n-body codes, a biomolecular analysis, and a chess problem.
How to write fast and clear parallel programs using algebra
Stiller, L. |
1992-10-01
An algebraic method for the design of efficient and easy to port codes for parallel machines is described. The method was applied to speed up and to clarify certain communication functions, n-body codes, a biomolecular analysis, and a chess problem.
Geometric and Algebraic Approaches in the Concept of Complex Numbers
ERIC Educational Resources Information Center
Panaoura, A.; Elia, I.; Gagatsis, A.; Giatilis, G.-P.
2006-01-01
This study explores pupils' performance and processes in tasks involving equations and inequalities of complex numbers requiring conversions from a geometric representation to an algebraic representation and conversions in the reverse direction, and also in complex numbers problem solving. Data were collected from 95 pupils of the final grade from…
Layout optimization with algebraic multigrid methods
NASA Technical Reports Server (NTRS)
Regler, Hans; Ruede, Ulrich
1993-01-01
Finding the optimal position for the individual cells (also called functional modules) on the chip surface is an important and difficult step in the design of integrated circuits. This paper deals with the problem of relative placement, that is the minimization of a quadratic functional with a large, sparse, positive definite system matrix. The basic optimization problem must be augmented by constraints to inhibit solutions where cells overlap. Besides classical iterative methods, based on conjugate gradients (CG), we show that algebraic multigrid methods (AMG) provide an interesting alternative. For moderately sized examples with about 10000 cells, AMG is already competitive with CG and is expected to be superior for larger problems. Besides the classical 'multiplicative' AMG algorithm where the levels are visited sequentially, we propose an 'additive' variant of AMG where levels may be treated in parallel and that is suitable as a preconditioner in the CG algorithm.
Words and possible words in early language acquisition.
Marchetto, Erika; Bonatti, Luca L
2013-11-01
In order to acquire language, infants must extract its building blocks-words-and master the rules governing their legal combinations from speech. These two problems are not independent, however: words also have internal structure. Thus, infants must extract two kinds of information from the same speech input. They must find the actual words of their language. Furthermore, they must identify its possible words, that is, the sequences of sounds that, being morphologically well formed, could be words. Here, we show that infants' sensitivity to possible words appears to be more primitive and fundamental than their ability to find actual words. We expose 12- and 18-month-old infants to an artificial language containing a conflict between statistically coherent and structurally coherent items. We show that 18-month-olds can extract possible words when the familiarization stream contains marks of segmentation, but cannot do so when the stream is continuous. Yet, they can find actual words from a continuous stream by computing statistical relationships among syllables. By contrast, 12-month-olds can find possible words when familiarized with a segmented stream, but seem unable to extract statistically coherent items from a continuous stream that contains minimal conflicts between statistical and structural information. These results suggest that sensitivity to word structure is in place earlier than the ability to analyze distributional information. The ability to compute nontrivial statistical relationships becomes fully effective relatively late in development, when infants have already acquired a considerable amount of linguistic knowledge. Thus, mechanisms for structure extraction that do not rely on extensive sampling of the input are likely to have a much larger role in language acquisition than general-purpose statistical abilities. PMID:24041871
Words and possible words in early language acquisition.
Marchetto, Erika; Bonatti, Luca L
2013-11-01
In order to acquire language, infants must extract its building blocks-words-and master the rules governing their legal combinations from speech. These two problems are not independent, however: words also have internal structure. Thus, infants must extract two kinds of information from the same speech input. They must find the actual words of their language. Furthermore, they must identify its possible words, that is, the sequences of sounds that, being morphologically well formed, could be words. Here, we show that infants' sensitivity to possible words appears to be more primitive and fundamental than their ability to find actual words. We expose 12- and 18-month-old infants to an artificial language containing a conflict between statistically coherent and structurally coherent items. We show that 18-month-olds can extract possible words when the familiarization stream contains marks of segmentation, but cannot do so when the stream is continuous. Yet, they can find actual words from a continuous stream by computing statistical relationships among syllables. By contrast, 12-month-olds can find possible words when familiarized with a segmented stream, but seem unable to extract statistically coherent items from a continuous stream that contains minimal conflicts between statistical and structural information. These results suggest that sensitivity to word structure is in place earlier than the ability to analyze distributional information. The ability to compute nontrivial statistical relationships becomes fully effective relatively late in development, when infants have already acquired a considerable amount of linguistic knowledge. Thus, mechanisms for structure extraction that do not rely on extensive sampling of the input are likely to have a much larger role in language acquisition than general-purpose statistical abilities.
Description of DASSL: a differential/algebraic system solver
Petzold, L.R.
1982-09-01
This paper describes a new code DASSL, for the numerical solution of implicit systems of differential/algebraic equations. These equations are written in the form F(t,y,y') = 0, and they can include systems which are substantially more complex than standard form ODE systems y' = f(t,y). Differential/algebraic equations occur in several diverse applications in the physical world. We outline the algorithms and strategies used in DASSL, and explain some of the features of the code. In addition, we outline briefly what needs to be done to solve a problem using DASSL.
Double Precision Differential/Algebraic Sensitivity Analysis Code
1995-06-02
DDASAC solves nonlinear initial-value problems involving stiff implicit systems of ordinary differential and algebraic equations. Purely algebraic nonlinear systems can also be solved, given an initial guess within the region of attraction of a solution. Options include automatic reconciliation of inconsistent initial states and derivatives, automatic initial step selection, direct concurrent parametric sensitivity analysis, and stopping at a prescribed value of any user-defined functional of the current solution vector. Local error control (in the max-normmore » or the 2-norm) is provided for the state vector and can include the sensitivities on request.« less
Exact solution of some linear matrix equations using algebraic methods
NASA Technical Reports Server (NTRS)
Djaferis, T. E.; Mitter, S. K.
1977-01-01
A study is done of solution methods for Linear Matrix Equations including Lyapunov's equation, using methods of modern algebra. The emphasis is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The action f sub BA is introduced a Basic Lemma is proven. The equation PA + BP = -C as well as the Lyapunov equation are analyzed. Algorithms are given for the solution of the Lyapunov and comment is given on its arithmetic complexity. The equation P - A'PA = Q is studied and numerical examples are given.
Parabosons, parafermions, and explicit representations of infinite-dimensional algebras
Stoilova, N. I.; Van der Jeugt, J.
2010-03-15
The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so({infinity}) and of the Lie superalgebra osp(1 vertical bar {infinity}). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labeled by certain infinite but stable Gelfand-Zetlin patterns, and the transformation of the basis is given explicitly. Alternatively, the basis vectors can be expressed as semi-standard Young tableaux.
Exact solution of some linear matrix equations using algebraic methods
NASA Technical Reports Server (NTRS)
Djaferis, T. E.; Mitter, S. K.
1979-01-01
Algebraic methods are used to construct the exact solution P of the linear matrix equation PA + BP = - C, where A, B, and C are matrices with real entries. The emphasis of this equation is on the use of finite algebraic procedures which are easily implemented on a digital computer and which lead to an explicit solution to the problem. The paper is divided into six sections which include the proof of the basic lemma, the Liapunov equation, and the computer implementation for the rational, integer and modular algorithms. Two numerical examples are given and the entire calculation process is depicted.
Parabosons, parafermions, and explicit representations of infinite-dimensional algebras
NASA Astrophysics Data System (ADS)
Stoilova, N. I.; van der Jeugt, J.
2010-03-01
The goal of this paper is to give an explicit construction of the Fock spaces of the parafermion and the paraboson algebra, for an infinite set of generators. This is equivalent to constructing certain unitary irreducible lowest weight representations of the (infinite rank) Lie algebra so(∞) and of the Lie superalgebra osp(1|∞). A complete solution to the problem is presented, in which the Fock spaces have basis vectors labeled by certain infinite but stable Gelfand-Zetlin patterns, and the transformation of the basis is given explicitly. Alternatively, the basis vectors can be expressed as semi-standard Young tableaux.
ERIC Educational Resources Information Center
Cantu, Virginia, Comp.; And Others
Prepared by bilingual teacher aide students, this glossary provides the Spanish translation of about 1,300 English words used in the bilingual classroom. Intended to serve as a handy reference for teachers, teacher aides, and students, the glossary can also be used in teacher training programs as a vocabulary builder for future bilingual teachers…
ERIC Educational Resources Information Center
Strauch-Nelson, Wendy
2007-01-01
Prompted by a parent's comment that indicated a desire for her elementary-age children to learn the elements and principles of design in their art class, the author set out to enrich her own understanding and appreciation of the language used in the art room. Looking at word origins helps students appreciate the significance of art and craft in…
Invertible linear transformations and the Lie algebras
NASA Astrophysics Data System (ADS)
Zhang, Yufeng; Tam, Honwah; Guo, Fukui
2008-07-01
With the help of invertible linear transformations and the known Lie algebras, a way to generate new Lie algebras is given. These Lie algebras obtained have a common feature, i.e. integrable couplings of solitary hierarchies could be obtained by using them, specially, the Hamiltonian structures of them could be worked out. Some ways to construct the loop algebras of the Lie algebras are presented. It follows that some various loop algebras are given. In addition, a few new Lie algebras are explicitly constructed in terms of the classification of Lie algebras proposed by Ma Wen-Xiu, which are bases for obtaining new Lie algebras by using invertible linear transformations. Finally, some solutions of a (2 + 1)-dimensional partial-differential equation hierarchy are obtained, whose Hamiltonian form-expressions are manifested by using the quadratic-form identity.
ERIC Educational Resources Information Center
Marchetto, Erika; Bonatti, Luca L.
2015-01-01
To achieve language proficiency, infants must find the building blocks of speech and master the rules governing their legal combinations. However, these problems are linked: words are also built according to rules. Here, we explored early morphosyntactic sensitivity by testing when and how infants could find either words or within-word structure…
Numerical algebraic geometry: a new perspective on gauge and string theories
NASA Astrophysics Data System (ADS)
Mehta, Dhagash; He, Yang-Hui; Hauensteine, Jonathan D.
2012-07-01
There is a rich interplay between algebraic geometry and string and gauge theories which has been recently aided immensely by advances in computational algebra. However, symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these shortcomings. The so-called `embarrassing parallelizability' allows us to solve many problems and extract physical information which elude symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.
NASA Technical Reports Server (NTRS)
Byrnes, C. I.
1980-01-01
It is noted that recent work by Kamen (1979) on the stability of half-plane digital filters shows that the problem of the existence of a feedback law also arises for other Banach algebras in applications. This situation calls for a realization theory and stabilizability criteria for systems defined over Banach for Frechet algebra A. Such a theory is developed here, with special emphasis placed on the construction of finitely generated realizations, the existence of coprime factorizations for T(s) defined over A, and the solvability of the quadratic optimal control problem and the associated algebraic Riccati equation over A.
Development of abstract mathematical reasoning: the case of algebra
Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja
2014-01-01
Algebra typically represents the students’ first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students’ ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16–17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15–16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students’ transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition. PMID:25228874
Development of abstract mathematical reasoning: the case of algebra.
Susac, Ana; Bubic, Andreja; Vrbanc, Andrija; Planinic, Maja
2014-01-01
Algebra typically represents the students' first encounter with abstract mathematical reasoning and it therefore causes significant difficulties for students who still reason concretely. The aim of the present study was to investigate the developmental trajectory of the students' ability to solve simple algebraic equations. 311 participants between the ages of 13 and 17 were given a computerized test of equation rearrangement. Equations consisted of an unknown and two other elements (numbers or letters), and the operations of multiplication/division. The obtained results showed that younger participants are less accurate and slower in solving equations with letters (symbols) than those with numbers. This difference disappeared for older participants (16-17 years), suggesting that they had reached an abstract reasoning level, at least for this simple task. A corresponding conclusion arises from the analysis of their strategies which suggests that younger participants mostly used concrete strategies such as inserting numbers, while older participants typically used more abstract, rule-based strategies. These results indicate that the development of algebraic thinking is a process which unfolds over a long period of time. In agreement with previous research, we can conclude that, on average, children at the age of 15-16 transition from using concrete to abstract strategies while solving the algebra problems addressed within the present study. A better understanding of the timing and speed of students' transition from concrete arithmetic reasoning to abstract algebraic reasoning might help in designing better curricula and teaching materials that would ease that transition.
A spatial operator algebra for manipulator modeling and control
NASA Technical Reports Server (NTRS)
Rodriguez, G.; Kreutz, Kenneth; Jain, Abhinandan
1989-01-01
A recently developed spatial operator algebra, useful for modeling, control, and trajectory design of manipulators is discussed. The elements of this algebra are linear operators whose domain and range spaces consist of forces, moments, velocities, and accelerations. The effect of these operators is equivalent to a spatial recursion along the span of a manipulator. Inversion of operators can be efficiently obtained via techniques of recursive filtering and smoothing. The operator algebra provides a high level framework for describing the dynamic and kinematic behavior of a manipulator and control and trajectory design algorithms. The interpretation of expressions within the algebraic framework leads to enhanced conceptual and physical understanding of manipulator dynamics and kinematics. Furthermore, implementable recursive algorithms can be immediately derived from the abstract operator expressions by inspection. Thus, the transition from an abstract problem formulation and solution to the detailed mechanizaton of specific algorithms is greatly simplified. The analytical formulation of the operator algebra, as well as its implementation in the Ada programming language are discussed.
Word Domain Disambiguation via Word Sense Disambiguation
Sanfilippo, Antonio P.; Tratz, Stephen C.; Gregory, Michelle L.
2006-06-04
Word subject domains have been widely used to improve the perform-ance of word sense disambiguation al-gorithms. However, comparatively little effort has been devoted so far to the disambiguation of word subject do-mains. The few existing approaches have focused on the development of al-gorithms specific to word domain dis-ambiguation. In this paper we explore an alternative approach where word domain disambiguation is achieved via word sense disambiguation. Our study shows that this approach yields very strong results, suggesting that word domain disambiguation can be ad-dressed in terms of word sense disam-biguation with no need for special purpose algorithms.
BRST charges for finite nonlinear algebras
NASA Astrophysics Data System (ADS)
Isaev, A. P.; Krivonos, S. O.; Ogievetsky, O. V.
2010-07-01
Some ingredients of the BRST construction for quantum Lie algebras are applied to a wider class of quadratic algebras of constraints. We build the BRST charge for a quantum Lie algebra with three generators and ghost-anti-ghosts commuting with constraints. We consider a one-parametric family of quadratic algebras with three generators and show that the BRST charge acquires the conventional form after a redefinition of ghosts. The modified ghosts form a quadratic algebra. The family possesses a nonlinear involution, which implies the existence of two independent BRST charges for each algebra in the family. These BRST charges anticommute and form a double BRST complex.
Some Remarks on Kite Pseudo Effect Algebras
NASA Astrophysics Data System (ADS)
Dvurečenskij, Anatolij; Holland, W. Charles
2014-05-01
Recently a new family of pseudo effect algebras, called kite pseudo effect algebras, was introduced. Such an algebra starts with a po-group G, a set I and with two bijections λ, ρ: I→ I. Using a clever construction on the ordinal sum of ( G +) I and ( G -) I , we can define a pseudo effect algebra which can be non-commutative even if G is an Abelian po-group. In the paper we give a characterization of subdirect product of subdirectly irreducible kite pseudo effect algebras, and we show that every kite pseudo effect algebra is an interval in a unital po-loop.
Operator product expansion algebra
Holland, Jan; Hollands, Stefan
2013-07-15
We establish conceptually important properties of the operator product expansion (OPE) in the context of perturbative, Euclidean φ{sup 4}-quantum field theory. First, we demonstrate, generalizing earlier results and techniques of hep-th/1105.3375, that the 3-point OPE,
Does "Word Coach" Coach Words?
ERIC Educational Resources Information Center
Cobb, Tom; Horst, Marlise
2011-01-01
This study reports on the design and testing of an integrated suite of vocabulary training games for Nintendo[TM] collectively designated "My Word Coach" (Ubisoft, 2008). The games' design is based on a wide range of learning research, from classic studies on recycling patterns to frequency studies of modern corpora. Its general usage and learning…
Constructing a parasupersymmetric Virasoro algebra
NASA Astrophysics Data System (ADS)
Kuwata, S.
2011-03-01
We construct a para SUSY Virasoro algebra by generalizing the ordinary fermion in SUSY Virasoro algebra (Ramond or Neveu-Schwarz algebra) to the parafermion. First, we obtain a polynomial relation (PR) between different-mode parafermion fi's by generalizing the corresponding single-mode PR to such that is invariant under the unitary transformation of fi (Green's condition). Differently from a usual context, where the Green's condition is imposed only on the defining relation of fi (degree three with respect to fi and fi†), we impose it on any degree of PR. For the case of order-two parafermion (the simplest case of para SUSY), we calculate a PR between the parasupercharge G0, the bosonic hamiltonian LB0 and parafermionic one LF0, although it is difficult to obtain a PR between G0 and the total hamiltonian L0 (= LB0 + LF0). Finally, we construct a para SUSY Virasoro algebra by generalizing L0 to the Ln's such that form a Virasoro algebra.
Word Fluency: A Task Analysis.
ERIC Educational Resources Information Center
Laine, Matti
It is suggested that models of human problem solving are useful in the analysis of word fluency (WF) test performance. In problem-solving terms, WF tasks would require the subject to define and clarify the conditions of the task (task acquisition), select and employ appropriate strategies, and monitor one's performance. In modern neuropsychology,…
People Considerations in Word Processing.
ERIC Educational Resources Information Center
Diamond, Marion L.
1984-01-01
Business educators preparing students for jobs in business and industry should become aware of the problems faced by workers in a typical large office environment. Word processor operators face many of the same problems as factory assembly line workers--lack of personalization, lack of incentive, and removal from the mainstream. (JOW)
Hidden algebra method (quasi-exact-solvability in quantum mechanics)
Turbiner, A. |
1996-02-01
A general introduction to quasi-exactly-solvable problems of quantum mechanics is presented. Main attention is given to multidimensional quasi-exactly-solvable and exactly-solvable Schroedinger operators. Exact-solvability of the Calogero and Sutherland {ital N}-body problems ass ociated with an existence of the hidden algebra {ital sl}{sub {ital N}} is discussed extensively. {copyright} {ital 1996 American Institute of Physics.}
Robot Control Based On Spatial-Operator Algebra
NASA Technical Reports Server (NTRS)
Rodriguez, Guillermo; Kreutz, Kenneth K.; Jain, Abhinandan
1992-01-01
Method for mathematical modeling and control of robotic manipulators based on spatial-operator algebra providing concise representation and simple, high-level theoretical frame-work for solution of kinematical and dynamical problems involving complicated temporal and spatial relationships. Recursive algorithms derived immediately from abstract spatial-operator expressions by inspection. Transition from abstract formulation through abstract solution to detailed implementation of specific algorithms to compute solution greatly simplified. Complicated dynamical problems like two cooperating robot arms solved more easily.
Using CAS to Solve Classical Mathematics Problems
ERIC Educational Resources Information Center
Burke, Maurice J.; Burroughs, Elizabeth A.
2009-01-01
Historically, calculus has displaced many algebraic methods for solving classical problems. This article illustrates an algebraic method for finding the zeros of polynomial functions that is closely related to Newton's method (devised in 1669, published in 1711), which is encountered in calculus. By exploring this problem, precalculus students…
Optimal Discretization Resolution in Algebraic Image Reconstruction
NASA Astrophysics Data System (ADS)
Sharif, Behzad; Kamalabadi, Farzad
2005-11-01
In this paper, we focus on data-limited tomographic imaging problems where the underlying linear inverse problem is ill-posed. A typical regularized reconstruction algorithm uses algebraic formulation with a predetermined discretization resolution. If the selected resolution is too low, we may loose useful details of the underlying image and if it is too high, the reconstruction will be unstable and the representation will fit irrelevant features. In this work, two approaches are introduced to address this issue. The first approach is using Mallow's CL method or generalized cross-validation. For each of the two methods, a joint estimator of regularization parameter and discretization resolution is proposed and their asymptotic optimality is investigated. The second approach is a Bayesian estimator of the model order using a complexity-penalizing prior. Numerical experiments focus on a space imaging application from a set of limited-angle tomographic observations.