Clustering via minimum volume ellipsoids

We propose minimum volume ellipsoids (MVE) clustering as an alternative clustering technique to k-means for data clusters with ellipsoidal shapes and explore its value and practicality. MVE clustering allocates data points into clusters in a way that minimizes the geometric mean of the volumes of each cluster’s covering ellipsoids. Motivations for this approach include its scale-invariance, its ability to handle asymmetric and unequal clusters, and our ability to formulate it as a mixed-integer semidefinite programming problem that can be solved to global optimality. We present some preliminary empirical results that illustrate MVE clustering as an appropriate method for clustering data from mixtures of “ellipsoidal” distributions and compare its performance with the k-means clustering algorithm as well as the MCLUST algorithm (which is based on a maximum likelihood EM algorithm) available in the statistical package R. Research of the first author was supported in part by a Discovery Grant from NSERC and a research grant from Faculty of Mathematics, University of Waterloo. Research of the second author was supported in part by a Discovery Grant from NSERC and a PREA from Ontario, Canada.     

Detection of functional states by the ‘LAMDA’ classification technique: application to a coagulation process in drinking water treatment

The present Note proposes a learning classification methodology to identify functional states on a coagulation process involved in drinking water treatment. In this work, we chose to carry out the supervised control of this process while using the LAMDA (Learning Algorithm for Multivariate Data Analysis) classification technique. The LAMDA classification technique proposes the interactive participation of the expert operator during the learning phase and in the optimisation of the classification. In this work, all information stemming from the environment process as well as expert knowledge has been aggregated and exploited. The application chosen for state identification is the Rocade drinking water treatment plant located at Marrakech, Morocco. To cite this article: B. Lamrini et al., C. R. Physique 6 (2005).     

Eliminating counterexamples: A Grade 8 student’s learning trajectory for contrapositive proving

This study uses a teaching experiment and retrospective analysis to develop a learning trajectory for improving a Grade 8 student’s ability to construct, critique, and validate contrapositive arguments. The study is predicated on the hypothesis that adolescents perform poorly on contrapositive reasoning tasks because they lack sufficient ways of justifying contrapositive argumentation as a viable mode of argumentation. By studying a student’s actions and comments as she develops, critiques, and validates not-the-conclusion-implies-the-conditions-are-impossible arguments for conditional claims, a promising learning trajectory for contrapositive argumentation is developed. The student’s learning trajectory demonstrates how a conception of contrapositive proving as eliminating counterexamples can be useful in developing, critiquing, and validating contrapositive arguments.     

Notions of equivalence through ratios: Students with and without learning disabilities

Students with learning disabilities (LD) specific to mathematics historically underperform in foundational content such as rational number equivalence. This study examined the strategy usage and multiplicative thinking of three third grade children (i.e., Bill, a child identified as having a learning disability specific to mathematics, Carl, a child labeled as low achieving in mathematics, and Albert, a child labeled as typically achieving) before, during, and after participating in tutoring sessions consisting of student-centered pedagogy and equivalence tasks presented through an underutilized interpretation of rational number: namely, the ratio interpretation. Constant comparison analysis of the children's work during the tutoring sessions as well as responses to tasks during two clinical interviews seemed to indicate that all three children increased their use of viable strategies, with notable differences in the sophistication of the strategies as well as the level of multiplicative thinking utilized before and after the ratio-based tutoring sessions. Yet, Bill's continued use of rudimentary strategies reflects a need for continued research to investigate why the use of such strategies persists and how supporting the development of more sophisticated strategies (especially among children with LD) can be achieved.     

Data-based spatial feedforward for over-actuated motion systems

For advanced motion systems there is an increasing demand for higher production throughput and accuracy. Traditionally, such systems are designed using a rigid-body design paradigm, which aims at designs with high stiffness. The alternative is to design a lightweight system and deal with the resulting flexibilities by over-actuation and over-sensing. This paper presents a data-based spatial feedforward method based on previous task trials, which aims at reducing the vibrations over the complete structure during motion. The proposed method is experimentally validated on an industrial prototype and compared to standard mass feedforward using rigid-body decoupling.     

Minimizing the total completion time in a single-machine scheduling problem with a learning effect

This paper introduces a new time-dependent learning effect model into a single-machine scheduling problem. The time-dependent learning effect means that the processing time of a job is assumed to be a function of total normal processing time of jobs scheduled in front of it. In most related studies, the actual job processing time is assumed to be a function of its scheduled position when the learning effect is considered in the scheduling problem. In this paper, the actual processing time of a job is assumed to be proportionate to the length and position of the already scheduled jobs. It shows that the addressed problem remains polynomially solvable for the objectives, i.e., minimization of the total completion time and minimization of the total weighted completion time. It also shows that the shortest processing time (SPT) rule provides the optimum sequence for the addressed problem.     

Fuzzy polynomial regression with fuzzy neural networks

Recently, fuzzy linear regression is considered by Mosleh et al. [1]. In this paper, a novel hybrid method based on fuzzy neural network for approximate fuzzy coefficients (parameters) of fuzzy polynomial regression models with fuzzy output and crisp inputs, is presented. Here a neural network is considered as a part of a large field called neural computing or soft computing. Moreover, in order to find the approximate parameters, a simple algorithm from the cost function of the fuzzy neural network is proposed. Finally, we illustrate our approach by some numerical examples.     

高级应用型人才培养模式及多媒体教学模式下的大学班主任工作随想

学风是校风的重要组成部分,是人才培养质量的重要保证。班主任在高校学风建设中起到非常重要的作用。班主任要有较高的责任心,用科学的班级管理方法,加强班风、学风建设,做好学生的引路人。     

A cycle for validating a learning progression illustrated with an example from the concept of function

A learning progression, or learning trajectory, describes the evolution of student thinking from early conceptions to the target understanding within a particular domain. As a complex theory of development, it requires conceptual and empirical support. In earlier work, we proposed a cycle for the validation of a learning progression with four steps: 1) Theory Development, 2) Examination of Empirical Recovery, 3) Comparison to Competing Models, and 4) Evaluation of Instructional Efficacy. A group of experts met to discuss the application of learning sciences to the design, use, and validation of classroom assessment. Learning progressions, learning trajectories, and how they can support classroom assessment were the main focuses. Revisions to the cycle were suggested. We describe the adapted cycle and illustrate how the first third of it has been applied towards the validation of a learning progression for the concept of function.     

Connecting multiplicative thinking and mathematical reasoning in the middle years

Mathematical reasoning and problem solving are recognised as essential 21st century skills. However, international assessments of mathematical literacy suggest these are areas of difficulty for many students. Evidenced-based learning trajectories that identify the key ideas and strategies needed to teach mathematics for understanding and support these important capacities over time are needed to support teachers and curriculum developers so that they do not have to rely solely on mathematics content knowledge. Given this goal and recent evidence to suggest a relationship between the development of multiplicative thinking and mathematical reasoning, this paper explores the processes involved in developing a single, integrated scale for multiplicative thinking and mathematical reasoning using data from a four-year design-based project to establish learning and assessment frameworks for algebraic, geometrical and statistical reasoning in the middle years of schooling.