The wholes that are greater than the sum of their parts: employing cooperative learning in mathematics teachers’ education

This paper presents a study on employing different types of cooperative learning (CL) settings in mathematics teacher education based on multiple research data. The study analyzes mechanisms in which CL contributes to the development of teacher knowledge of three kinds: subject matter knowledge, pedagogical content knowledge, and curricular content knowledge. Additionally, it reflects on interactions between different kinds of teachers’ knowledge in the process of development. Two wholes, which are greater than the sum of their parts, are analyzed in this paper. The first is a course whose design combines different CL settings, hence enhances different mechanisms for teachers’ professional development. The second whole, teachers’ “collaborative mind,” is presented as an outcome of the first. To exemplify possible contributions of employing CL in teacher education, this paper first zooms in on the structure of one particular course for mathematics teachers and then focuses on one particular mathematical activity.     

Unregularized online learning algorithms with general loss functions

In this paper, we consider unregularized online learning algorithms in a Reproducing Kernel Hilbert Space (RKHS). Firstly, we derive explicit convergence rates of the unregularized online learning algorithms for classification associated with a general α-activating loss (see Definition 1 below). Our results extend and refine the results in [30] for the least square loss and the recent result [3] for the loss function with a Lipschitz-continuous gradient. Moreover, we establish a very general condition on the step sizes which guarantees the convergence of the last iterate of such algorithms. Secondly, we establish, for the first time, the convergence of the unregularized pairwise learning algorithm with a general loss function and derive explicit rates under the assumption of polynomially decaying step sizes. Concrete examples are used to illustrate our main results. The main techniques are tools from convex analysis, refined inequalities of Gaussian averages [5], and an induction approach.     

Issues in theorizing mathematics learning and teaching: A contrast between learning through activity and DNR research programs

By continuing a contrast with the DNR research program, begun in Harel and Koichu (2010), I discuss several important issues with respect to teaching and learning mathematics that have emerged from our research program which studies learning that occurs through students’ mathematical activity and indicate issues of complementarity between DNR and our research program. I make distinctions about what we mean by inquiring into the mechanisms of conceptual learning and how it differs from work that elucidates steps in the development of a mathematical concept. I argue that the construct of disequilibrium is neither necessary nor sufficient to explain mathematics conceptual learning. I describe an emerging approach to instruction aimed at particular mathematical understandings that fosters reinvention of mathematical concepts without depending on students’ success solving novel problems.     

Fuzzy extensions of bargaining sets and their existence in cooperative fuzzy games

Recently, the concept of classical bargaining set given by Aumann and Maschler in 1964 has been extended to fuzzy bargaining set. In this paper, we give a modification to correct some weakness of this extension. We also extend the concept of the Mas-Colell's bargaining set (the other major type of bargaining sets) to its corresponding fuzzy bargaining set. Our main effort is to prove existence theorems for these two types of fuzzy bargaining sets. We will also give necessary and sufficient conditions for these bargaining sets to coincide with the Aubin Core in a continuous superadditive cooperative fuzzy game which has a crisp maximal coalition of maximum excess at each payoff vector. We show that both Aumann-Maschler and Mas-Colell fuzzy bargaining sets of a continuous convex cooperative fuzzy game coincide with its Aubin core.     

Combining latent learning with dynamic programming in the modular anticipatory classifier system

Learning Classifier Systems (LCS) are rule based Reinforcement Learning (RL) systems which use a generalization capability. In this paper, we highlight the differences between two kinds of LCSs. Some are used to directly perform RL while others latently learn a model of the interactions between the agent and its environment. Such a model can be used to speed up the core RL process. Thus, these two kinds of learning processes are complementary. We show here how the notion of generalization differs depending on whether the system anticipates (like Anticipatory Classifier System (ACS) and Yet Another Classifier System (YACS)) or not (like XCS). Moreover, we show some limitations of the formalism common to ACS and YACS, and propose a new system, called Modular Anticipatory Classifier System (MACS), which allows the latent learning process to take advantage of new regularities. We describe how the model can be used to perform active exploration and how this exploration may be aggregated with the policy resulting from the reinforcement learning process. The different algorithms are validated experimentally and some limitations in presence of uncertainties are highlighted.     

具有饱和项的互惠系统的分歧与稳定性

运用谱分析和分歧理论的方法,在齐次Dirichlet边界条件下,对具有饱和项的互惠系统的非负定态解的分歧及其稳定性进行研究.一方面,分别以生长率作为分歧参数,讨论了发自半平凡解的分歧;另一方面,以两物种的生长率作为分歧参数,利用Liapunov-Schmidt过程,研究了在二重特征值处的分歧;同时判定了这些分歧解的稳定性.     

The role of informal learning spaces in enhancing student engagement with mathematical sciences

By helping create a shared, supportive, learning community, the creative use of custom-designed spaces outside the classroom has a major impact on student engagement. The intention is to create spaces that promote peer interaction within and across year groups, encourage closer working relationships between staff and students and support specific coursework activities – particularly group work. Such spaces make better use of time since students are motivated to stay and work during long gaps in their timetable, can provide a sense of ‘home’ within the institution and can lead to a cohesive community of practice. In this paper, we describe how this has been achieved and currently delivered in Mathematics at Sheffield Hallam University and provide some detailed analysis of the student usage of the space.     

Approximating the least core value and least core of cooperative games with supermodular costs

We study the approximation of the least core value and the least core of supermodular cost cooperative games. We provide a framework for approximation based on oracles that approximately determine maximally violated constraints. This framework yields a 3-approximation algorithm for computing the least core value of supermodular cost cooperative games, and a polynomial-time algorithm for computing a cost allocation in the 2-approximate least core of these games. This approximation framework extends naturally to submodular profit cooperative games. For scheduling games, a special class of supermodular cost cooperative games, we give a fully polynomial-time approximation scheme for computing the least core value. For matroid profit games, a special class of submodular profit cooperative games, we give exact polynomial-time algorithms for computing the least core value as well as a least core cost allocation.     

Vendor Management Inventory with consignment contracts and the benefits of cooperative advertising

Most of the cooperative advertising literature has focused on studying the effects of such programs considering marketing variables. This paper integrates production and inventory management with pricing and advertising considerations to assess the effects of cooperative advertising programs in bilateral monopolies. We consider a supply chain where a Vendor Managed Inventory (VMI) along with a consignment contract is implemented to coordinate the chain. We develop and solve a differential model for two games. The first one is a benchmark scenario where no cooperative advertising is offered, while the manufacturer offers the cooperative program in the second game. The main results show that cooperative advertising programs, usually considered as successful marketing initiatives, can be very difficult to implement in a supply chain undertaking a VMI policy with a consignment contract, in which operations and marketing interface is taken into account. A cooperative program mainly hurts the manufacturer’s profits, and can be profit-Pareto-improving only in a few cases. Although the retailer is generally willing to receive a support from the manufacturer, she can opt for a non-cooperative program when the largest part of the supply chain profits goes to the manufacturer. We developed several special cases to strengthen our findings.     

Single-machine scheduling with learning and forgetting effects

Scheduling with learning effects has been widely studied in the past decade. With the increasingly moving toward shorter product cycle times in many production lines, workers in this changeable environment must constantly learn new skill and technology. As a result, the forgetting effect might occur in these situations. In this paper, we propose a model with the consideration of both the learning and forgetting effects. We show some single-machine problems remain polynomially solvable under the proposed model.     

On a cooperative advertising model for a supply chain with one manufacturer and one retailer

, , ,  and  recently studied a game-theoretic model for cooperative advertising in a supply chain consisting of one manufacturer and one retailer. However, the sales-volume (demand) function considered in this model can become negative for some values of the decision variables, and in fact, this does happen for the proposed Stackelberg and Nash equilibrium solutions. Yue et al. (2006) acknowledge the negativity problem and suggest two constraints to fix it; however, they do not incorporate these constraints into their mathematical analysis. In this paper, we show that the results obtained by analyzing the advertising model under the constraints suggested by Yue et al. can differ significantly from those obtained in the previous papers.     

Oligopolistic markets with leadership,cooperative followers and a multivalued inverse demand function

We consider an oligopolistic market with a leader, finitely many followers and a multivalued inverse demand function. By assuming that there exist cooperative behaviors of followers, we introduce the notions of Stackelberg Cournot equilibria for markets with nontransferable utilities and transferable utilities. We shall prove their existence theorems in our models.     

Single-machine scheduling with a general sum-of-actual-processing-times-based and job-position-based learning effect

In this paper, we bring into the scheduling field a general learning effect model where the actual processing time of a job is not only a general function of the total actual processing times of the jobs already processed, but also a general function of the job’s scheduled position. We show that the makespan minimization problem and the sum of the kth power of completion times minimization problem can be solved in polynomial time, respectively. We also show that the total weighted completion time minimization problem and the maximum lateness minimization problem can be solved in polynomial time under certain conditions.     

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.     

Scheduling with deteriorating jobs and learning effects

This paper studies a single machine scheduling problem simultaneously with deteriorating jobs and learning effects. The objectives are to minimize the makespan and the number of tardy jobs, respectively. Two polynomial time algorithms are proposed to solve these problems optimally.     

Investment using evolutionary learning methods and technical rules

In this paper, I propose a genetic learning approach to generate technical trading systems for stock timing. The most informative technical indicators are selected from a set of almost 5000 signals by a multi-objective genetic algorithm with variable string length. Successively, these signals are combined into a unique trading signal by a learning method. I test the expert weighting solution obtained by the plurality voting committee, the Bayesian model averaging and Boosting procedures with data from the S&P 500 Composite Index, in three market phases, up-trend, down-trend and sideways-movements, covering the period 2000–2006. Computational results indicate that the near-optimal set of rules varies among market phases but presents stable results and is able to reduce or eliminate losses in down-trend periods.     

Sum-fuzzy implementation of a choice function using artificial learning procedure with fixed fraction

In one if his paper Luo transformed the problem of sum-fuzzy rationality into artificial learning procedure and gave an algorithm which used the learning rule of perception. This paper extends the Luo method for finding a sum-fuzzy implementation of a choice function and offers an algorithm based on the artificial learning procedure with fixed fraction. We also present a concrete example which uses this algorithm.