Expanding Notions of “Learning Trajectories” in Mathematics Education

Over the past 20 years learning trajectories and learning progressions have gained prominence in mathematics and science education research. However, use of these representations ranges widely in breadth and depth, often depending on from what discipline they emerge and the type of learning they intend to characterize. Learning trajectories research has spanned from studies of individual student learning of a single concept to trajectories covering a full set of content standards across grade bands. In this article, we discuss important theoretical assumptions that implicitly guide the development and use of learning trajectories and progressions in mathematics education. We argue that diverse theoretical conceptualizations of what it means for a student to “learn” mathematics necessarily both constrains and amplifies what a particular learning trajectory can capture about the development of students’ knowledge.     

An evidential reasoning based consensus model for multiple attribute group decision analysis problems with interval-valued group consensus requirements

With the aim of modeling multiple attribute group decision analysis problems with group consensus (GC) requirements, a GC based evidential reasoning approach and further an attribute weight based feedback model are sequentially developed based on an evidential reasoning (ER) approach. In real situations, however, giving precise (crisp) assessments for alternatives is often too restrictive and difficult for experts, due to incompleteness or lack of information. Experts may also find it difficult to give appropriate assessments on specific attributes, due to limitation or lack of knowledge, experience and provided data about the problem domain. In this paper, an ER based consensus model (ERCM) is proposed to deal with these situations, in which experts’ assessments are interval-valued rather than precise. Correspondingly, predefined interval-valued GC (IGC) requirements need to be reached after group analysis and discussion within specified times. Also, the process of reaching IGC is accelerated by a feedback mechanism including identification rules at three levels, consisting of the attribute, alternative and global levels, and a suggestion rule. Particularly, recommendations on assessments in the suggestion rule are constructed based on recommendations on their lower and upper bounds detected by the identification rule at a specific level. A preferentially developed industry selection problem is solved by the ERCM to demonstrate its detailed implementation process, validity, and applicability.     

Pre‐Service Teachers and Mathematics: The Impact of Service‐Learning on Teacher Preparation

In spite of repeated reform efforts, there is research and data that suggest that teachers lack the needed knowledge to successfully teach elementary mathematics. Some argue that teachers lack the needed content knowledge while other argue that a lack of confidence and practice are impacting teachers’ ability to successful teach mathematics. As a result of these issues, this paper looks at the impact of a service‐learning experience on pre‐service teachers’ confidence and preparation in the area of mathematics. The service‐learning experience had a number of intended and unintended outcomes. As a result of the service‐learning experience, pre‐service teachers noted increased understanding, innovation and confidence in the area of mathematics.     

A Learning Progression for Elementary Students’ Functional Thinking

In this article we advance characterizations of and supports for elementary students’ progress in generalizing and representing functional relationships as part of a comprehensive approach to early algebra. Our learning progressions approach to early algebra research involves the coordination of a curricular framework and progression, an instructional sequence, written assessments, and levels of sophistication describing students’ algebraic thinking. After detailing this approach, we focus on what we have learned about the development of students’ abilities to generalize and represent functional relationships in a grades 3–5 early algebra intervention by sharing the levels of responses we observed in students’ written work over time. We found that the sophistication of students’ responses increased over the course of the intervention from recursive patterning to correspondence and in some cases covariation relationships between variables. Students’ responses at times differed by the particular tasks that were posed. We discuss implications for research and practice.     

Location-based techniques for the synergy approximation in combinatorial transportation auctions

The use of combinatorial auctions for the procurement of transportation services is investigated in this paper. We focus on the carrier viewpoint who is interested in submitting to the auction a selected bundle of loads that avoids some of the empty movement of the vehicles in his transportation network and that increases his profits. For this purpose we develop an optimization approach based on the use of the location techniques, usually used in the context of facility planning. Mathematically, this means maximizing the synergy among the bundle’s auctioned loads from one side and between the auctioned and the pre-existing loads from the other side. We show the validity of our approach by using first an illustrative example and then by applying it to solve a real-life problem related to a logistics company installed in the Arabic Gulf region.     

Young Children's Composition of Geometric Figures: A Learning Trajectory

The purpose of this research is to chart the mathematical actions-on-objects young children use to compose geometric shapes. The ultimate goal is the creation of a hypothetical learning trajectory based on previous research, as well as instrumentation to assess levels of learning along the developmental progression underlying the trajectory. We tested both the developmental progression and the instrument through a series of studies, including formative studies (including action research by 8 teachers) and a summative study involving 72 children ages 3 to 7 years. Results provide strong support for the validity of the developmental progression's levels and suggest that children move through these levels of thinking in developing the ability to compose 2-dimensional figures. From lack of competence in composing geometric shapes, they gain abilities to combine shapes-initially through trial and error and gradually by attributes-into pictures, and finally synthesize combinations of shapes into new shapes (composite shapes).     

Locally averaged Bayesian Dirichlet metrics for learning the structure and the parameters of Bayesian networks

The marginal likelihood of the data computed using Bayesian score metrics is at the core of score+search methods when learning Bayesian networks from data. However, common formulations of those Bayesian score metrics rely on free parameters which are hard to assess. Recent theoretical and experimental works have also shown that the commonly employed BDe score metric is strongly biased by the particular assignments of its free parameter known as the equivalent sample size. This sensitivity means that poor choices of this parameter lead to inferred BN models whose structure and parameters do not properly represent the distribution generating the data even for large sample sizes. In this paper we argue that the problem is that the BDe metric is based on assumptions about the BN model parameters distribution assumed to generate the data which are too strict and do not hold in real settings. To overcome this issue we introduce here an approach that tries to marginalize the meta-parameter locally, aiming to embrace a wider set of assumptions about these parameters. It is shown experimentally that this approach offers a robust performance, as good as that of the standard BDe metric with an optimum selection of its free parameter and, in consequence, this method prevents the choice of wrong settings for this widely applied Bayesian score metric.     

Improvement heuristic for the flow-shop scheduling problem: An adaptive-learning approach

We propose an improvement-heuristic approach for the general flow-shop problem (n/m/C_max) based on the idea of adaptive learning. The approach employs a one-pass heuristic to give a good starting solution in the search space and uses a weight parameter to perturb the data of the original problem to obtain improved solutions. The weights are then adjusted employing a learning strategy which involves reinforcement and backtracking. The learning is similar to that in neural networks. The random perturbation allows a non-deterministic local search. We apply the improvement-heuristic approach in conjunction with three well-known heuristics in the literature, namely, Palmer’s Slope Index, CDS and NEH. We test our approach on several benchmark problem sets including Taillard’s, Carlier’s, Heller’s and Reeves’. We compare our results to the best-known upper-bound solutions and find that for many problems we match the best-known upper bound. For one problem we discover a new upper bound.     

A Study of STEM Assessments in Engineering,Science, and Mathematics for Elementary and Middle School Students

The purpose of this study was to develop, scale, and validate assessments in engineering, science, and mathematics with grade appropriate items that were sensitive to the curriculum developed by teachers. The use of item response theory to assess item functioning was a focus of the study. The work is part of a larger project focused on increasing student learning in science, technology, engineering, and mathematics (STEM)‐related areas in grades 4–8 through an engineering design‐based, integrated approach to STEM instruction and assessment. The fact that the assessments are available to school districts at no cost, and represent psychometrically sound instruments that are sensitive to STEM‐oriented curriculum, offers schools an important tool for gauging students' understanding of engineering, science, and mathematics concepts.     

Neuroevolution strategies for episodic reinforcement learning

Because of their convincing performance, there is a growing interest in using evolutionary algorithms for reinforcement learning. We propose learning of neural network policies by the covariance matrix adaptation evolution strategy (CMA-ES), a randomized variable-metric search algorithm for continuous optimization. We argue that this approach, which we refer to as CMA Neuroevolution Strategy (CMA-NeuroES), is ideally suited for reinforcement learning, in particular because it is based on ranking policies (and therefore robust against noise), efficiently detects correlations between parameters, and infers a search direction from scalar reinforcement signals. We evaluate the CMA-NeuroES on five different (Markovian and non-Markovian) variants of the common pole balancing problem. The results are compared to those described in a recent study covering several RL algorithms, and the CMA-NeuroES shows the overall best performance.     

The Metaphysics of Mixed Inferences: Problems with Functionalist Accounts of Alethic Pluralism

Alethic pluralists argue truth is a metaphysically robust higher-order property that is multiply realized by a set of diverse and domain-specific subvening alethic properties. The higher-order truth property legitimizes mixed inferences and accounts for a univocal truth predicate. Absent of this higher-order property, pluralists lack an account of the validity of mixed inferences and an adequate semantics for the truth predicate and thereby appear forced to abandon the central tenets of alethic pluralism. I argue the use of many-valued logics to support pluralism fails to address the pluralist’s metaphysical problem regarding mixed inferences and mixed truth functional connectives. The high degree of heterogeneity of the alethic realizers (unlike the realizers for pain) challenges the plausibility of a single higher-order functional property. A functional property with such a heterogeneous base cannot be projectable at a theoretically significant level. The problem with mixed inferences and truth functions is but one symptom of the deeper projectability problem.     

Residual algorithm for large-scale positive definite generalized eigenvalue problems

In the positive definite case, the extreme generalized eigenvalues can be obtained by solving a suitable nonlinear system of equations. In this work, we adapt and study the use of recently developed low-cost derivative-free residual schemes for nonlinear systems, to solve large-scale generalized eigenvalue problems. We demonstrate the effectiveness of our approach on some standard test problems, and also on a problem associated with the vibration analysis of large structures. In our numerical results we use preconditioning strategies based on incomplete factorizations, and we compare with and without preconditioning with a well-known available package.     

An iterative approach to solve a nonlinear moving boundary problem describing the solvent diffusion within glassy polymers

This paper consists in studying a mathematical model of solvent diffusion through the glassy polymers as a one-dimensional moving boundary problem with kinetic undercooling. We establish an iterative variable time-step method based on a nonstandard finite difference (NSFD) scheme to solve the considered moving boundary problem. The monotonicity and positivity of the numerical solution are proved. The numerical approach is investigated for three test problems composed of constant and inconstant diffusion coefficients for different values of parameters to demonstrate the validity and ability of the method.     

Fuzzy modelling through logic optimization

This study proposes a new logic-driven approach to the development of fuzzy models. We introduce a two-phase design process realizing adaptive logic processing in the form of structural and parametric optimization. By recognizing the fundamental links between binary (two-valued) and fuzzy (multi-valued) logic, effective structural learning is achieved through the use of well-established methods of Boolean minimization encountered in digital systems. This blueprint structure is then refined by adjusting connections of fuzzy neurons, helping to capture the numeric details of the target system’s behavior. The introduced structure along with the learning mechanisms helps achieve high accuracy and interpretability (transparency) of the resulting model.     

Multi-objective Meta-heuristics for the Traveling Salesman Problem with Profits

We introduce and test a new approach for the bi-objective routing problem known as the traveling salesman problem with profits. This problem deals with the optimization of two conflicting objectives: the minimization of the tour length and the maximization of the collected profits. This problem has been studied in the form of a single objective problem, where either the two objectives have been combined or one of the objectives has been treated as a constraint. The purpose of our study is to find solutions to this problem using the notion of Pareto optimality, i.e. by searching for efficient solutions and constructing an efficient frontier. We have developed an ejection chain local search and combined it with a multi-objective evolutionary algorithm which is used to generate diversified starting solutions in the objective space. We apply our hybrid meta-heuristic to synthetic data sets and demonstrate its effectiveness by comparing our results with a procedure that employs one of the best single-objective approaches.      

基于二元语义的犹豫模糊语言决策方法

当决策者在给出语言评价信息而表示出犹豫时,决策信息更适合用犹豫模糊语言术语集表达。为了减少语言决策过程中信息的丢失,得到较精准的评价结果,本文提出基于二元语义的犹豫模糊语言决策方法。首先定义了犹豫模糊二元语义集、犹豫模糊二元语义集的均值函数、方差函数及其集结算子,然后用集结算子求出各方案的综合评价值,通过犹豫模糊二元语义的均值函数和方差函数确定方案排序。最后通过实例说明了该方法的实用性和有效性。     

Solution Bounds for Elliptic Partial Differential Equations via Feynman-Kac Representation

We transform suitable smooth functions into hard bounds for the solution to boundary value and obstacle problems for elliptic partial differential equations based on the probabilistic Feynman-Kac representation. Unlike standard approximate solutions, hard solution bounds are intended to limit the location of the solution, possibly to a large extent, and, thus, have the potential to be very useful information. Our approach requires two main steps. First, the violation of sufficient conditions is quantified for the test function to be a hard bounding function. After extracting those violation terms from the Feynman-Kac representation, it remains to deal with a boundary value problem with constant input data. Although the probabilistic Feynman-Kac representation is employed, the resulting numerical method is deterministic without the need for sophisticated probabilistic numerical methods, such as sample paths generation of reflected diffusion processes. Throughout this article, we provide numerical examples to illustrate the effectiveness of the proposed method.