A sentence generator for testing parsers

A fast algorithm is given to produce a small set of short sentences from a context free grammar such that each production of the grammar is used at least once. The sentences are useful for testing parsing programs and for debugging grammars (finding errors in a grammar which causes it to specify some language other than the one intended). Some experimental results from using the sentences to test some automatically generated simpleLR(1) parsers are also given.     

Capturing and characterizing students’ strategic algebraic reasoning through cognitively demanding tasks with focus on representations

In mathematics education, it is important to assess valued practices such as problem solving and communication. Yet, often we assess students based on correct solutions over their problem solving strategies—strategies that can uncover important mathematical understanding. In this article, we first present a framework of competencies required for strategic reasoning to solve cognitively demanding algebra tasks and assessment tools to capture evidence of these competencies. Then, we qualitatively describe characteristics of student reasoning for various performance levels (low, medium, and high) of eighth-grade students, focusing on generating and interpreting algebraic representations. We argue this analysis allows a more comprehensive and complex perspective of student understanding. Our findings lay groundwork to investigate the continuum of algebraic understanding, and may help educators identify specific areas of students’ strength and weakness when solving cognitively demanding tasks.     

An adaptive anisotropic perfectly matched layer method for 3-D time harmonic electromagnetic scattering problems

We develop an anisotropic perfectly matched layer (PML) method for solving the time harmonic electromagnetic scattering problems in which the PML coordinate stretching is performed only in one direction outside a cuboid domain. The PML parameters such as the thickness of the layer and the absorbing medium property are determined through sharp a posteriori error estimates. Combined with the adaptive finite element method, the proposed adaptive anisotropic PML method provides a complete numerical strategy to solve the scattering problem in the framework of FEM which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the choice of the thickness of the PML layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.     

An adaptive perfectly matched layer technique for 3-D time-harmonic electromagnetic scattering problems

An adaptive perfectly matched layer (PML) technique for solving the time harmonic electromagnetic scattering problems is developed. The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. Combined with the adaptive finite element method, the adaptive PML technique provides a complete numerical strategy to solve the scattering problem in the framework of FEM which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorbing layer. Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.

     

Student teachers’ concept definitions of area and their understanding about two-dimensionality of area

We examine what kind of concept definitions of area a group of Finnish primary and lower secondary student teachers (N = 82) use, and how the quality of the definitions is associated with the participants’ success in seven exercises involving area. We are especially interested in how the understanding of the two-dimensionality of area appears in the participants’ responses. Only six student teachers were able to give a mathematically precise and correct definition of area. Altogether 26 participants defined it as ‘the size of a figure’ and 20 respondents required that a figure must be bounded. Further, 22 of them associated area with a formula or an example and eight respondents gave an incorrect or nonsensical definition. On average, student teachers master rather well the area formulae of a circle and a rectangle but already the relationship between the surface area of a cube and its volume is less commonly perceived. Most student teachers associate the area of an irregular domain with the method of exhaustion but clearly fewer of them acknowledge the difference between the area and an approximation of it. Surprisingly, there is only a weak Spearman correlation between the participants’ scores in the test exercises and the qualitatively ordered categories of concept definitions.     

Stretching,sliding and strategy: indexicality in algebraic explanations

In this study, we describe the linguistic expression of strategy in explanations of algebraic procedures. Stating the steps of an algebraic procedure does not require a student to indicate the relationship between different mathematical actions, but describing algebraic strategy does. The coordinated nature of strategic proficiency suggests that linguistic forms known as indexical language, “pointing words” that link speech to context, may be fundamental resources for expressing this type of competence. A class of first-year university mathematics students developed a habit of reporting procedures that we consider a speech genre. The classroom genre emphasised procedural explanations, but when students expressed strategic competence, they often relied on indexical language. Indexical verbs of motion like slide and drop proved to be a particularly efficient means of expressing algebraic strategies. This informal speech style extended the communicative capacity of the classroom speech genre, and allowed classmates to participate better in strategic mathematical reasoning.     

Mapping and development of intuitive proportional thinking

The purpose of this study was two-fold. First, to find out students’ informal understanding of proportional problems, and discuss their solution strategies. Second, to investigate how the intuitions developed by students influence their strategies to solve proportional problems. To this end, we interviewed 16 students in Grades 4 and 5, while they were solving proportional problems. It was found that students intuitively used the unit-rate strategy indicating an attempt to transfer the knowledge resulted by their experience with solving simple multiplicative problems. Fourth and fifth graders tended to shift from the unit-rate strategy to other strategies if there was no easy way to calculate the unit-value directly from the context of the problems. Since fifth graders were more comfortable than fourth graders in calculating the unit-value, they felt less the need to invent other solution strategies.     

The nature and development of experts’ strategy flexibility for solving equations

Largely absent from the emerging literature on flexibility is a consideration of experts’ flexibility. Do experts exhibit strategy flexibility, as one might assume? If so, how do experts perceive that this capacity developed in themselves? Do experts feel that flexibility is an important instructional outcome in school mathematics? In this paper, we describe results from several interviews with experts to explore strategy flexibility for solving equations. We conducted interviews with eight content experts, where we asked a number of questions about flexibility and also engaged the experts in problem solving. Our analysis indicates that the experts that were interviewed did exhibit strategy flexibility in the domain of linear equation solving, but they did not consistently select the most efficient method for solving a given equation. However, regardless of whether these experts used the best method on a given problem, they nevertheless showed an awareness of and an appreciation of efficient and elegant problem solutions. The experts that we spoke to were capable of making subtle judgments about the most appropriate strategy for a given problem, based on factors including mental and rapid testing of strategies, the problem solver’s goals (e.g., efficiency, error-free execution, elegance) and familiarity with a given problem type. Implications for future research on flexibility and on mathematics instruction are discussed.     

Design and research potential of interactive textbooks: the case of fractions

The transition from classical to electronic textbooks seems to be a logical step in the digitization advancing worldwide. However, developing an e-book ought to be more than digitizing text: key features of computer-based learning environments such as interactive exercises, adaptive demands, or automatic feedback should be integrated to take advantage of the digitization. The “ALICE:fractions” project aims at designing and evaluating an interactive mathematics textbook for introducing fractions in the classroom. It is based on research in mathematics education and includes the elements just mentioned. This paper describes the electronic textbook’s implementation and its theoretical background. Moreover, it introduces aspects that allow further information on learning processes to be gleaned. As an example, time on task is regarded in a study with 6th graders who used the electronic textbook in the classroom. Linear mixed models revealed a negative effect of time on task on task success. The effect was moderated by exercise difficulty and slightly by student competence: the effect was less pronounced in difficult exercises and for low-achieving students, whereas for high-achieving students or in easier exercises, the effect intensified.     

A hybrid tabu search/branch & bound approach to solving the generalized assignment problem

A new approach for solving the generalized assignment problem (GAP) is proposed that combines the exact branch & bound approach with the heuristic strategy of tabu search (TS) to produce a hybrid algorithm for solving GAP. The algorithm described uses commercial software to solve sub-problems generated by the TS guiding strategy. The TS approach makes use of the concept of referent domain optimisation and introduces novel add/drop strategies. In addition, the linear programming relaxation of GAP that forms part of the branch & bound approach is itself helpful in suggesting which variables might take binary values. Computational results on benchmark test instances are presented and compared with results obtained by the standard branch & bound approach and also several other heuristic approaches from the literature. The results show the new algorithm performs competitively against the alternatives and is able to find some new best solutions for several benchmark instances.     

Global optimization problems and domain reduction strategies

In this paper we discuss domain reduction strategies for global optimization problems with a nonconvex objective function over a bounded convex feasible region. After introducing a standard domain reduction and its iterated version, we will introduce a new reduction strategy. Under mild assumptions, we will prove the equivalence between the new domain reduction and the iterated version of the standard one, allowing a new interpretation of the latter and a new way of computing it. Finally, we prove that any “reasonable” domain reduction strategy is independent of the order by which variables are processed.     

AgentGeom: a multiagent system for pedagogical support in geometric proof problems

This paper aims, first, to describe the fundamental characteristics and workings of the AgentGeom artificial tutorial system, which is designed to help students develop knowledge and skills related to problem solving, mathematical proof in geometry, and the use of mathematical language. Following this, we indicate the manner in which a secondary school student can appropriate these abilities through interactions with the system. Our system uses strategic messages of the agent tutor in an argumentative process that collaborates with a student in the construction of a proof.     

Computer-Assisted,Semi-Programmed,and Teaching Assistant-Led Instruction in General Chemistry

The purpose of this study was to investigate computer-assisted instruction and semi-programmed instruction as replacements for traditional recitation/discussion in general chemistry. Student performance on problem solving and student attitudes were investigated. The study also examined the effect of formal reasoning ability as a covariate. The data indicated that for simple exercises, all methods of teaching recitation/discussion sections were equally effective, but for more difficult exercises, the semi-programmed instruction was most effective. Students had more positive attitudes when using the semi-programmed instruction than when attending traditional recitation/discussion sections.     

Adaptive smoothing algorithms for nonsmooth composite convex minimization

We propose an adaptive smoothing algorithm based on Nesterov’s smoothing technique in Nesterov (Math Prog 103(1):127–152, 2005) for solving “fully” nonsmooth composite convex optimization problems. Our method combines both Nesterov’s accelerated proximal gradient scheme and a new homotopy strategy for smoothness parameter. By an appropriate choice of smoothing functions, we develop a new algorithm that has the \(\mathcal {O}\left( \frac{1}{\varepsilon }\right) \)-worst-case iteration-complexity while preserves the same complexity-per-iteration as in Nesterov’s method and allows one to automatically update the smoothness parameter at each iteration. Then, we customize our algorithm to solve four special cases that cover various applications. We also specify our algorithm to solve constrained convex optimization problems and show its convergence guarantee on a primal sequence of iterates. We demonstrate our algorithm through three numerical examples and compare it with other related algorithms.     

An Adaptive Uniaxial Perfectly Matched Layer Method for Time-Harmonic Scattering Problems

The uniaxial perfectly matched layer (PML) method uses rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in deal- ing with problems involving anisotropic scatterers.In this paper an adaptive uniaxial PML technique for solving the time harmonic Helmholtz scattering problem is devel- oped.The PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates.The adaptive finite element method based on a posteriori error estimate is proposed to solve the PML equa- tion which produces automatically a coarse mesh size away from the fixed domain and thus makes the total computational costs insensitive to the thickness of the PML absorb- ing layer.Numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method.In particular,it is demonstrated that the PML layer can be chosen as close to one wave-length from the scatterer and still yields good accuracy and efficiency in approximating the far fields.     

The Role of Interactions Between Student and Classroom Context in Developing Adaptive Self‐Efficacy in One Sixth‐Grade Mathematics Classroom

Research and theory suggest several instructional practices that could enhance student self‐efficacy. However, little is known about the ways these instructional practices interact with individual students to create opportunities or challenges for developing adaptive self‐efficacy. In this study, we focused on two sources of efficacy, mastery experiences, and social persuasion, and examined how these sources were structured for three students with different levels of mathematics achievement and self‐efficacy within a sixth‐grade mathematics classroom. Analyses within each case showed that each student experienced success and received social persuasion differently. On the other hand, analyses across the cases suggest that not only the amount but also the form (i.e., with and without assistance) of successful experiences and the type of performances (e.g., stating definitions, explaining solution procedures, sharing problem‐solving strategies, and making comments on others' ideas) through which the student experienced mastery may have played important role in developing self‐efficacy. Consistently, the amount and form of teacher feedback was different for each focal student. Examining these differences provide insight into each student's self‐efficacy assessed over the course of the study as well as the kind of support each student needed to develop adaptive self‐efficacy.     

A Formal Language for Cryptographic Protocol Requirements

In this paper we present a formal language for specifying and reasoning about cryptographic protocol requirements. We give sets of requirements for key distribution protocols and for key agreement protocols in that language. We look at a key agreement protocol due to Aziz and Diffie that might meet those requirements and show how to specify it in the language of the NRL Protocol Analyzer. We also show how to map our formal requirements to the language of the NRL Protocol Analyzer and use the Analyzer to show that the protocol meets those requirements. In other words, we use the Analyzer to assess the validity of the formulae that make up the requirements in models of the protocol. Our analysis reveals an implicit assumption about implementations of the protocol and reveals subtleties in the kinds of requirements one might specify for similar protocols.     

Teaching OR: automatic evaluation for linear programming modelling

4OR - Learning how to model a problem described in natural language as a linear program requires students to practice using various and numerous exercises. Moreover, immediate feedback on the...     

Open-source tools for dynamical analysis of Liley's mean-field cortex model

Mean-field models of the mammalian cortex treat this part of the brain as a two-dimensional excitable medium. The electrical potentials, generated by the excitatory and inhibitory neuron populations, are described by nonlinear, coupled, partial differential equations that are known to generate complicated spatio-temporal behaviour. We focus on the model by Liley et al. (Network: Computation in Neural Systems 13 (2002) 67–113). Several reductions of this model have been studied in detail, but a direct analysis of its spatio-temporal dynamics has, to the best of our knowledge, never been attempted before. Here, we describe the implementation of implicit time-stepping of the model and the tangent linear model, and solving for equilibria and time-periodic solutions, using the open-source library PETSc. By using domain decomposition for parallelization, and iterative solving of linear problems, the code is capable of parsing some dynamics of a macroscopic slice of cortical tissue with a sub-millimetre resolution.     

The Effect of Journal Writing on Achievement in and Attitudes Toward Mathematics

A teaching experiment was conducted to investigate the effect of journal writing on achievement in and attitudes toward mathematics. Achievement variables included conceptual understanding, procedural knowledge, problem solving, mathematics school achievement, and mathematical communication. Subjects were selected from first intermediate students (11–13 years) attending the International College, Beirut, Lebanon, where either English or French is the language of mathematics instruction. The journal-writing (JW) group received the same mathematics instruction as the no-journal-writing (NJW) group, except that the JW group engaged in prompted journal writing for 7 to 10 minutes at the end of each class period, three times a week, for 12 weeks. The NJW group engaged in exercises during the same period. The results of ANCOVA suggest that journal writing has a positive impact on conceptual understanding, procedural knowledge, and mathematical communication but not on problem solving, school mathematics achievement, and attitudes toward mathematics. Gender, language of instruction, mathematics achievement level, and writing achievement level failed to interact with journal writing. Student responses to a questionnaire indicated that students found journal writing to have both cognitive and affective benefits.