Trees and Keislers problem

We give a new negative solution to Keisler's problem regarding Skolem functions and elementary extensions. In contrast to existing ad hoc solutions due to Payne, Knight, and Lachlan, our solution uses well-known models. Received: 20 September 1999 / Published online: 21 March 2001     

A neural networks based model of inverse hysteresis

A novel and simple approach based on transformation using neural networks is proposed in this paper to model the inverse behavior of hysteresis. In this approach, a continuous transformation is used to construct an elementary inverse hysteresis operator (EIHO), which can extract the change tendency of inverse hysteresis. Then based on the EIHO, an expanded input space is constructed to transform the multi-valued mapping of inverse hysteresis into a one-to-one mapping. Based on the constructed expanded input space, a neural network is employed to approximate the inverse hysteresis. Both experiment and simulation are implemented to validate the effectiveness of the proposed approach. These results indicate that the proposed approach has derived satisfactory modeling performance.     

The finite-to-finite strand of a learning progression for the concept of function: A research synthesis and cognitive analysis

In this paper we report validation efforts around the finite-to-finite strand of a provisional learning progression (LP) for the concept of function. We regard an LP as an empirically-verified account of how student understandings form over time and in response to instruction. The finite-to-finite strand of the LP was informed by literature on students’ thinking and learning related to functions as well as the Algebra Project’s curricular approach, which is designed for students who are traditionally underserved by mathematics education. Developing and validating an LP is a multi-step, cyclic process. Here we report on one step in this process, an item and response analysis. Data sources include 680 students’ responses to 13 multipart computer-delivered tasks. Results suggest that revisions to the items, associated scoring rubrics, and in some instances the LP are warranted. We illustrate this task, rubric, and LP revision process through an item analysis for a selected task.     

Efficient elementary and restricted non-elementary route pricing

Column generation is involved in the current most efficient approaches to routing problems. Set partitioning formulations model routing problems by considering all possible routes and selecting a subset that visits all customers. These formulations often produce tight lower bounds and require column generation for their pricing step. The bounds in the resulting branch-and-price are tighter when elementary routes are considered, but this approach leads to a more difficult pricing problem. Balancing the pricing with route relaxations has become crucial for the efficiency of the branch-and-price for routing problems. Recently, the ng-routes relaxation was proposed as a compromise between elementary and non-elementary routes. The ng-routes are non-elementary routes with the restriction that when following a customer, the route is not allowed to visit another customer that was visited before if they belong to a dynamically computed set. The larger the size of these sets, the closer the ng-route is to an elementary route. This work presents an efficient pricing algorithm for ng-routes and extends this algorithm for elementary routes. Therefore, we address the Shortest Path Problem with Resource Constraint (SPPRC) and the Elementary Shortest Path Problem with Resource Constraint (ESPPRC). The proposed algorithm combines the Decremental State-Space Relaxation technique (DSSR) with completion bounds. We apply this algorithm for the Generalized Vehicle Routing Problem (GVRP) and for the Capacitated Vehicle Routing Problem (CVRP), demonstrating that it is able to price elementary routes for instances up to 200 customers, a result that doubles the size of the ESPPRC instances solved to date.     

Students’ coordination of lower and higher dimensional units in the context of constructing and evaluating sums of consecutive whole numbers

This paper examines how three eighth grade students coordinated lower and higher dimensional units (e.g., composite units and pairs) in the context of constructing a formula for evaluating sums of consecutive whole numbers while solving combinatorics problems (e.g., 1 + 2 + ⋯ + 15 = (16 × 15)/2). The data is drawn from the beginning of an 8-month teaching experiment. The findings from the study include: (1) a framework for understanding how students coordinate lower and higher dimensional units; (2) identification of key learning that occurred as students made the transition between solving two kinds of combinatorics problems; and (3) identification of the links between the way students’ coordinated lower and higher dimensional units and their evaluation of sums of consecutive whole numbers. Implications for research and teaching are considered.     

From “You have to have three numbers and a plus sign” to “It’s the exact same thing”: K–1 students learn to think relationally about equations

This research shares progressions in thinking about equations and the equal sign observed in ten students who took part in an early algebra classroom intervention across Kindergarten and first grade. We report on data from task-based interviews conducted prior to the intervention and at the conclusion of each school year that elicited students’ interpretations of the equal sign and equations of various forms. We found at the beginning of the intervention that most students viewed the equal sign as an operational symbol and did not accept many equations forms as valid. By the end of first grade, almost all students described the symbol as indicating the equivalence of two amounts and were much more successful interpreting and working with equations in a variety of forms. The progressions we observed align with those of other researchers and provide evidence that very young students can learn to reason flexibly about equations.     

李三系的Frattini子系

李三系是从黎曼对称空间产生的三元运算的代数系统,近年来备受数学家们的重视.针对李三系的Frattini子系和基本李三系的问题进行了研究,给出了Frattini子系和基本李三系的一些性质,并证明了李三系的非嵌入定理,同时得到了幂零李三系是基本李三系的一个充要条件.     

Examination of the dynamic software-supported learning environment in data analysis

The Fatih Project in Turkey has improved software in mathematics teaching such as data analysis software. As a result, the need to inquire into the efficiency of computer-supported learning environments has emerged. This study aims to examine the effect of dynamic data analysis software-supported learning environments on secondary school students’ achievement and attitude. The research method employs a quasi-experimental design with a pre-test, post-test control group. Basic topics related to data analysis were introduced through dynamic statistics software in the experimental group while the students were taught with the help of smart boards, course books and exercises in the control group. Data were collected with an achievement test, attitude scale and semi-structured interviews. Also, interviews were conducted with four students from the experimental group in order to get more detailed information from students. The data gained in the study were analysed both quantitatively and qualitatively. The findings revealed that statistics teaching through statistics software is more efficient than the one with the traditional method on achievement and attitudes. In accordance with this result, it is suggested that computer-supported statistics software should be used in statistics teaching.     

Evaluating students’ conceptual and procedural understanding of sampling distributions

Introductory statistics courses, which are important in preparing students for their daily lives, generally derive inferential statistics from informal knowledge. In this transition process, sampling distributions have an important place, yet research has shown that students often have difficulties with this concept. In order to increase their understanding of sampling distributions, students should have a strong conceptual foundation that is balanced with procedural knowledge. To address this issue, this study was designed to examine the relationship between college students’ procedural and conceptual knowledge of sampling distributions. With this aim in mind, an achievement test consisting of two sections – procedural and conceptual knowledge – was prepared. In answering the questions related to procedural knowledge, the participants were more successful in identifying the relationship between standard deviation of a population and sample means. However, they lacked theoretical knowledge about statements that they had heard or knew intuitively. Simulation activities provided in statistics courses may support students in developing their conceptual understanding in this regard.