Algebraic Tasks Research Articles

Systematic Mathematical Errors and Cognitive Load

The hypothesis that the intrinsic nature of algebraic bracket tasks causes an uneven distribution of cognitive load during computation was tested in three experiments with grades 8 and 9 students. In Experiment 1, students were given problems which required two successive brackets to be expanded; each bracket required two operations (computations) to be completed. It was discovered that more errors were made during the calculation of the second bracket than the first, and more errors were made during the second operation than the first operation within each bracket. Verbal protocols collected in Experiment 2 indicated that most errors were caused by failures in working memory rather than poorly learned rules. In Experiment 3, a dual-task methodology showed that the cognitive nature of brackets affected working memory performance. It was concluded that the cognitive load experienced by problem solvers on these tasks varied across operations and caused the observed error pattern.

Symbolic Maximum Likelihood Estimation with Mathematica

Mathematica is a symbolic programming language that empowers the user to undertake complicated algebraic tasks. One such task is the derivation of maximum likelihood estimators, demonstrably an important topic in statistics at both the research and expository level. In this paper, a Mathematica package is provided that contains a function entitled SuperLog. This function utilises pattern-matching code that enhances Mathematica's ability to simplify expressions involving the natural logarithm of a product of algebraic terms. This enhancement to Mathematica's functionality can be of particular benefit for maximum likelihood estimation.

Selfexcitation of a MDF‐System with Coulomb friction, varying normal forces and impacts

AbstractThe investigation is focused on nonlinear oscillations of a beam structure with unilateral contact to a sliding surface. Analytical solutions for each state (stack, slip, free oscillation, impact) are used to reduce the non‐smoothness of the system to an algebraic task. The numerical results are compared with experimental data.

Understanding of Logical Necessity: Developmental Antecedents and Cognitive Consequences

Does abstract reasoning develop naturally, and does instruction contribute to its development? In an attempt to answer these questions, this article specifically focuses on effects of prolonged instruction on the development of abstract deductive reasoning and, more specifically, on the development of understanding of logical necessity. It was hypothesized that instructional emphasis on the metalevel of deduction within a knowledge domain can amplify the development of deductive reasoning both within and across this domain. The article presents 2 studies that examine the development of understanding of logical necessity in algebraic and verbal deductive reasoning. In the first study, algebraic and verbal reasoning tasks were administered to 450 younger and older adolescents selected across different instructional settings in England and in Russia. In the second study, algebraic and verbal reasoning tasks were administered to 287 Russian younger and older adolescents selected across different instructional settings. The results support the hypothesis, indicating that prolonged instruction with an emphasis on the metalevel of algebraic deduction contributes to the development of understanding of logical necessity in both algebraic and verbal deductive reasoning. Findings also suggest that many adolescents do not develop an understanding of logical necessity naturally.

Understanding of Logical Necessity: Developmental Antecedents and Cognitive Consequences

Does abstract reasoning develop naturally, and does instruction contribute to its development? In an attempt to answer these questions, this article specifically focuses on effects of prolonged instruction on the development of abstract deductive reasoning and, more specifically, on the development of understanding of logical necessity. It was hypothesized that instructional emphasis on the metalevel of deduction within a knowledge domain can amplify the development of deductive reasoning both within and across this domain. The article presents 2 studies that examine the development of understanding of logical necessity in algebraic and verbal deductive reasoning. In the first study, algebraic and verbal reasoning tasks were administered to 450 younger and older adolescents selected across different instructional settings in England and in Russia. In the second study, algebraic and verbal reasoning tasks were administered to 287 Russian younger and older adolescents selected across different instructional settings. The results support the hypothesis, indicating that prolonged instruction with an emphasis on the metalevel of algebraic deduction contributes to the development of understanding of logical necessity in both algebraic and verbal deductive reasoning. Findings also suggest that many adolescents do not develop an understanding of logical necessity naturally.

Coupled cells with internal symmetry: II. Direct products

We continue the study of arrays of coupled identical cells that possess both global and internal symmetries, begun in part I. Here we concentrate on the `direct product' case, for which the symmetry group of the system decomposes as the direct product of the internal group and the global group . Again, the main aim is to find general existence conditions for symmetry-breaking steady-state and Hopf bifurcations by reducing the problem to known results for systems with symmetry or separately. Unlike the wreath product case, the theory makes extensive use of the representation theory of compact Lie groups. Again the central algebraic task is to classify axial and -axial subgroups of the direct product and to relate them to axial and -axial subgroups of the two groups and . We demonstrate how the results lead to efficient classification by studying both steady state and Hopf bifurcation in rings of coupled cells, where and . In particular we show that for Hopf bifurcation the case n = 4 modulo 4 is exceptional, by exhibiting two extra types of solution that occur only for those values of n.

Quantum methods with Mathematica

The first quantum mechanics text published that ties directly into a computer algebra system, this book exploits Mathematica(r) throughout for symbolic, numeric, and graphical computing. It is a work designed for computer interaction in an upper-division undergraduate or first-year graduate quantum mechanics course. It is also a toolbox for the practicing physicist seeking to automate a variety of algebraic and numerical tasks with the computer. The book is divided into two parts: Systems in One Dimension and Quantum Dynamics. Part I emphasizes topics from a first-year course on quantum mechanics, while Part II includes more advanced topics. Although the text requires some familiarity with Mathematica, appendices are provided for gaining experience with the software and are referenced throughout the book. The text is task-oriented and integrated with numerous problems and exercises, with hints for working on the computer. James M. Feagin is a Professor of Physics at California State University, Fullerton. He was educated at Georgia Tech and the University of North Carolina, Chapel Hill, where he received a Ph.D. in theoretical physics in 1980. He is a Fellow of the Alexander von Humboldt Foundation and has served as visiting Professor at the University of Freiburg, Germany. Feagin is the author of numerous articles on collision physics and the dynamics of few-body systems. He has given a number of invited talks and hosted workshops on incorporating computers into the physics curriculum and is presently helping to introduce computing into the Introductory University Physics Project (IUPP) sponsored by the American Institute of Physics.

Applications of optical Boolean matrix operations to graph theory

The transition from optical numerical matrix algebra to optical Boolean matrix algebra is explored in detail. All important Boolean matrix algebra tasks can be performed optically. Quantitative measurement is replaced by a simple light-or-no-light decision, something optics can do well. The parallelism advantage of optics becomes greater as the matrix size increases. As an illustration of utility, we consider graph theory.

Balanced realizations via gradient flow techniques

The task of finding a class of balanced minimal realizations is shown to be equivalent to finding limiting solutions of certain gradient flow differential equations. By viewing such algebraic tasks in the context of calculus, they are amenable to analog computational solutions, or parallel processing machines, perhaps even neural networks. The convergence rates of the differential equations is exponential, and consequentially convergence is rapid and numerical stability properties are attractive.

The arithmetic connection

From test and interview data obtained during an investigation into Grade 10 students' conceptions of algebraic generalisation and justification, we have extracted evidence of the extent to which these students have coordinated the “worlds” of arithmetic and algebra, and can move freely between them. The data show more dissociation than we expected, even among students who were successful at standard algebraic tasks. Conceiving algebra as “generalised arithmetic” may obscure the many genuine obstacles that the learner has to overcome in moving from fluent performance in arithmetic to fluent performance in algebra while achieving and maintaining a smooth coordination of both modes of action.

Basic algebra revisited: a study with 14-year-olds

This paper reports the results obtained with a group of 24 14-year-old pupils when presented with sets of algebra tasks by the Leeds Modelling System, LMS. Four months later, these same pupils were given a comparable paper-and-pencil test and detailed interviews. A comparison between these sets of results is presented. The results obtained on the paper-and-pencil test and the interviews were consistent, and show that the pupils had some profound misunderstandings of algebraic notation. Further, from the interviews it is possible to determine classes of strategies some pupils were using, namely: — searching for solutions (i.e. substituting); — applying a “global” rule, such as collecting ALL the numbers on one side, whether or not they were coefficients. Moreover, this work further demonstrates the importance of interviews to interpret curious protocols.

An Attempt to Understand Students' Understanding of Basic Algebra*

This paper reports the results obtained with a group of 24 14‐year‐old students when presented with a set of algebra tasks by the Leeds Modelling System, LMS. These same students were given a comparable paper‐and‐pencil test and detailed interviews some four months later. The latter studies uncovered several kinds of student misunderstandings that LMS had not detected. Some students had profound misunderstandings of algebraic notation: Others used strategies such as substituting numbers for variables until the equation balanced. Additionally, it appears that the student errors fall into several distinct classes: namely, manipulative, parsing, clerical, and “random.”LMS and its rule database have been enhanced as the result of this experiment, and LMS is now able to diagnose the majority of the errors encountered in this experiment. Finally, the paper gives a process‐oriented explanation for student errors, and re‐examines related work in cognitive modelling in the light of the types of student errors reported in this experiment. Misgeneralization is a mechanism suggested to explain some of the mal‐rules noted in this study.

Factoring Puzzles

Factor puzzles are a clever technique for building readiness to tackle the complex algebraic task of factoring trinomials.