An inquiry-oriented approach to undergraduate mathematics

To improve undergraduate mathematics learning, teachers need to recognize and value characteristics of classroom learning environments that contribute to powerful student learning. The broad goal of this special issue is to share such characteristics and the theoretical and empirical grounding for an innovative approach in differential equations called the Inquiry Oriented Differential Equations (IO-DE) project. We use the IO-DE project as a case example of how undergraduate mathematics can build on theoretical and instructional advances initiated at the K-12 level to create and sustain learning environments for powerful student learning at the undergraduate level. In addition to providing an overview of the five articles in this special issue, we highlight the theoretical background for the IO-DE project and provide a summary of two quantitative studies done to assess the effectiveness of the IO-DE project on student learning.     

Learning to represent,representing to learn

This study explores how students learn to create, discuss, and reason with representations to solve problems. A summer school algebra class for seventh and eighth graders provided opportunities for students to create and use representations as problem-solving tools. This case study follows the learning trajectories of three boys. Two of the three boys had been low-achievers in their previous math classes, and one was a high achiever. Analysis of all three boys’ written work reveals how their representations became more sophisticated over time. Their small group interactions while problem-solving also show changes in how they communicated and reasoned with representations. For these boys, representation functioned as a learning practice. Through constructing and reasoning with representations, the boys were able to engage in generalizing and justifying claims, discuss quadratic growth, and collaborate and persist in problem-solving. Negotiating different student-constructed representations of a problem also gave them opportunities to act with agency, as they made choices and judgments about the validity of the different perspectives. These findings have implications for the importance of giving all students access to mathematics through representations, with representational thinking serving as a central disciplinary practice and as a learning practice that supports further mathematics learning.     

Extending the participatory learning paradigm to include source credibility

We provide an overview of the participatory learning paradigm (PLP) and discuss the importance of the acceptance function in determining which observations are used for learning. We introduce a formal model that uses this (PLP) We then extend this model in two directions. First, we consider situations in which we have incomplete observations, we only have observations about a subset of the variables of interest. Next we extend this model to allow for the inclusion in the learning process of information about the learning agents belief about the credibility of the source of the learning experience. Here we distinguish between the content of a learning experience and the source of the experience. We provide a means to allow the learning agents belief about the credibility of the source to determine the effect of the content. Furthermore we suggest a method to allow the modification of agents belief about the credibility of the source to also be part of the learning process.     

New approaches to statistical learning theory

We present new tools from probability theory that can be applied to the analysis of learning algorithms. These tools allow to derive new bounds on the generalization performance of learning algorithms and to propose alternative measures of the complexity of the learning task, which in turn can be used to derive new learning algorithms.     

A market-based computational approach to collaborative organizational learning

We draw upon the concepts of knowledge market, organizational tacit knowledge, credit assignment, and single-loop learning in proposing a market-based conceptual model for collaborative organizational learning. Our proposed model is characterized by the local competition among seller agents and the global collaboration among winner agents in forming a plan, through a chain of ‘upstream–downstream’ working relationship, for task accomplishment. This feature is achieved through three closely coupled processes: the expert selection process, the capital reallocation process, and the plan formation process. Our model is intended for multiple-step learning environment in which each task consists of a sequence of single-step learning tasks. Learning at the global level is the result of a sequence of nested single-loop learning at the local level.     

A social perspective on technology-enhanced mathematical learning: from collaboration to performance

This paper documents both developments in the technologies used to promote learning mathematics and the influence on research of social theories of learning, through reference to the activities of the International Commission on Mathematical Instruction (ICMI), and argues that these changes provide opportunity for the reconceptualization of our understanding of mathematical learning. Firstly, changes in technology are traced from discipline-specific computer-based software through to Web 2.0-based learning tools. Secondly, the increasing influence of social theories of learning on mathematics education research is reviewed by examining the prevalence of papers and presentations, which acknowledge the role of social interaction in learning, at ICMI conferences over the past 20 years. Finally, it is argued that the confluence of these developments means that it is necessary to re-examine what it means to learn and do mathematics and proposes that it is now possible to view learning mathematics as an activity that is performed rather than passively acquired.     

浅谈在高等数学教学中如何将基本概念形象化

高等数学中的基本概念具有高度抽象性，是教师讲授的难点，也是学生学习的重点，而基本概念的掌握是学生学习理解高等数学的关键，也是培养学生学习兴趣的前提．本总结了多种方法，将基本概念形象化变为可“听”，可“看”，可“摸”，降低了基本概念的教与学难度，提高了学生学好高等数学的兴趣。     

A modified approach to team-based learning in linear algebra courses

This paper documents the author’s adaptation of team-based learning (TBL), an active learning pedagogy developed by Larry Michaelsen and others, in the linear algebra classroom. The paper discusses the standard components of TBL and the necessary changes to those components for the needs of the course in question. There is also an empirically controlled analysis of the effects of TBL on the student learning experience in the first year of TBL use.     

Vector-valued reproducing kernel Banach spaces with applications to multi-task learning

Motivated by multi-task machine learning with Banach spaces, we propose the notion of vector-valued reproducing kernel Banach spaces (RKBSs). Basic properties of the spaces and the associated reproducing kernels are investigated. We also present feature map constructions and several concrete examples of vector-valued RKBSs. The theory is then applied to multi-task machine learning. Especially, the representer theorem and characterization equations for the minimizer of regularized learning schemes in vector-valued RKBSs are established.     

Bayesian network approach to multinomial parameter learning using data and expert judgments

One of the hardest challenges in building a realistic Bayesian Network (BN) model is to construct the node probability tables (NPTs). Even with a fixed predefined model structure and very large amounts of relevant data, machine learning methods do not consistently achieve great accuracy compared to the ground truth when learning the NPT entries (parameters). Hence, it is widely believed that incorporating expert judgments can improve the learning process. We present a multinomial parameter learning method, which can easily incorporate both expert judgments and data during the parameter learning process. This method uses an auxiliary BN model to learn the parameters of a given BN. The auxiliary BN contains continuous variables and the parameter estimation amounts to updating these variables using an iterative discretization technique. The expert judgments are provided in the form of constraints on parameters divided into two categories: linear inequality constraints and approximate equality constraints. The method is evaluated with experiments based on a number of well-known sample BN models (such as Asia, Alarm and Hailfinder) as well as a real-world software defects prediction BN model. Empirically, the new method achieves much greater learning accuracy (compared to both state-of-the-art machine learning techniques and directly competing methods) with much less data. For example, in the software defects BN for a sample size of 20 (which would be considered difficult to collect in practice) when a small number of real expert constraints are provided, our method achieves a level of accuracy in parameter estimation that can only be matched by other methods with much larger sample sizes (320 samples required for the standard machine learning method, and 105 for the directly competing method with constraints).     

Learning to align organizational design and data

The relationship between organizational learning and organizational design is explored. In particular, we examine the information processing aspects of organizational learning as they apply to a two-valued decision making task and the relation of such aspects to organizational structure. Our primary contribution is to extend Carley's (1992) model of this process. The original model assumes that all data input into the decision making processes are of equal importance or weight in determining the correct overall organizational decision. The extension described here allows for the more natural situation of non-uniform weights of evidence. Further extensions to the model are also discussed. Such organizational learning performance measures provide an interesting framework for analyzing the recent trend towards flatter organizational structures. This research suggests that flatter structures are not always better, but rather that data environment, ultimate performance goals, and relative need for speed in learning can be used to form a contingency model for choosing organizational structure.     

Using virtual sample generation to build up management knowledge in the early manufacturing stages

Since only few examples can be obtained in the early stages in a manufacturing system and that fewer exemplars usually lead to a lower learning accuracy, this research uses intervalized kernel methods of Density Estimation to improve the small-data-set learning. Used techniques include the Intervalization Process to improve the kernel density estimation and virtual sample generation to produce extra information for expediting the learning. Results obtained from the provided example, using a back-propagation neural network as the learning tool, show that this unique approach is an effective method of scheduling knowledge creation for a system in the early stages.     

A framework for using learning theories to inform ‘growth mindset’ activities

ABSTRACT

The social psychology theory of fixed and growth mindsets offers one reason for observed underachievement in science, technology, engineering and mathematics (STEM), particularly for students who have previously excelled in these disciplines. Fixed mindset beliefs are linked to behaviours that can lead to avoiding challenges and reduced learning, such as concealing a lack of understanding to retain an image of being ‘smart’. The potential impact of a growth mindset on STEM achievement, particularly for minority and low-household-income students, resulted in calls to develop interventions that encourage growth mindsets and discourage fixed mindsets. However, education interventions are influenced by the educator's understanding of how learning occurs. A framework to show how activities based on different learning theories may encourage growth mindsets or (unintentionally) encourage fixed mindsets can guide the developers of growth mindset interventions. We present such a framework in six tables relating to key areas associated with growth and fixed mindsets: dealing with challenges, persistence, effort, praise, the success of others and learning goals. Each table gives examples of learning activities that may encourage growth or fixed mindsets, fitting with each of four key learning theories: behaviourism, constructivism, communities of practice and connectivism.     

A perspective on artificial intelligence: Learning to learn

The classical approach to the acquisition of knowledge in artificial intelligence has been to program the intelligence into the machine in the form of specific rules for the application of the knowledge: expert systems. Unfortunately, the amount of time and resources required to program an expert system with sufficient knowledge for non-trivial problem-solving is prohibitively large. An alternative approach is to allow the machine tolearn the rules based upon trial-and-error interaction with the environment, much as humans do. This will require endowing the machine with a sophisticated set of sensors for the perception of the external world, the ability to generate trial actions based upon this perceived information, and a dynamic evaluation policy to allow it to measure the effectiveness of its trial actions and modify its repertoire accordingly. The principles underlying this paradigm, known ascollective learning systems theory, have already been applied to sophisticated gaming problems, demonstrating robust learning and dynamic adaptivity.The fundamental building block of a collective learning system is thelearning cell, which may be embedded in a massively parallel, hierarchical data communications network. Such a network comprising 100 million learning cells will approach the intelligence capacity of the human cortex. In the not-too-distant future, it may be possible to build a race of robotic slaves to perform a wide variety of tasks in our culture. This goal, while irresistibly attractive, is most certainly fraught with severe social, political, moral, and economic difficulties.This paper was given as an invited talk on the 12th Symposium on Operations Research, University of Passau, September 1987.     

Spaces to discuss mathematics: communities of practice on an online discussion board

In theories of learning that adopt a situated stance to knowledge the notion of identity is vital; how learners position themselves in relation to, and are mutually positioned by, the situation within which they are learning will have a strong bearing on the learning outcomes. One of the challenges for learning mathematics in school is that learners position themselves, and are positioned, as pupils rather than as mathematicians. This paper focuses on discussion boards designed for secondary school mathematics students, and we use Wenger's (1998) model of communities of practice, building on earlier work by the authors (Back and Pratt 2007; Pratt and Kelly 2007) in which ‘idealised communities’ are constructed and used, to consider a case study of one participant who engages in developing his identity as a mathematician doing mathematics, as well his identity as a learner and a teacher of mathematics.     

Learning to prove: from examples to general statements

In this paper we describe a method for teaching students to prove some mathematical statements independently, by using specially designed auxiliary assignments. The assignments are designed as homework problems and can be adapted for online learning. We illustrate our method using examples from calculus and differential equations.     

Work allocation to stations with varying learning slopes and without buffers

In this paper, we address the problem of allocating the work elements, belonging to the products of a lot, to the stations of an assembly line so as to minimize the makespan. The lots that are processed on the assembly line are characterized by a low overall demand for each product. There is no buffer permitted in between the stations, and the line operates under learning. In particular, the stations’ learning slopes are assumed to be different. We present a procedure to determine the optimal assignments of the workload to the stations under learning variability and show that it considerably affects these assignments.     

LTC: A latent tree approach to classification

Latent tree models were proposed as a class of models for unsupervised learning, and have been applied to various problems such as clustering and density estimation. In this paper, we study the usefulness of latent tree models in another paradigm, namely supervised learning. We propose a novel generative classifier called latent tree classifier (LTC). An LTC represents each class-conditional distribution of attributes using a latent tree model, and uses Bayes rule to make prediction. Latent tree models can capture complex relationship among attributes. Therefore, LTC is able to approximate the true distribution behind data well and thus achieves good classification accuracy. We present an algorithm for learning LTC and empirically evaluate it on an extensive collection of UCI data. The results show that LTC compares favorably to the state-of-the-art in terms of classification accuracy. We also demonstrate that LTC can reveal underlying concepts and discover interesting subgroups within each class.     

Learning to rank: a ROC-based graph-theoretic approach

This note summarizes the main results presented in the author’s Ph.D. thesis, supervised by Luc Boullart and Bernard De Baets. The thesis was defended on 14th October 2008 at Universiteit Gent. It is written in English and available for download at . The work deals with preference learning, with emphasis on the ranking and ordinal regression machine learning settings and their connections to decision theory. Based on receiver operator characteristics analysis and graph theory, new performance measures are proposed to evaluate this type of models, and new algorithms are presented to compute and optimize these performance measures efficiently. Furthermore, the relationship with other settings like pairwise preference learning and multi-class classification is discussed.     

Incorporating the learning effect into data envelopment analysis to measure MSW recycling performance

The effect of organizational learning, which results in continuous improvement of organizational performance over time, has been widely discussed. The cumulative learning effect may form as a source of intellectual capital. Thus far, the static data envelopment analysis (DEA) model has not been used to examine the longitudinal learning effect. Therefore, a two-stage approach is developed together with the estimation of a latent learning effect using time-series data; the estimated learning effect is then used as an input in the DEA Slacks-Based Measure (SBM) model. The proposed DEA SBM model can be used to investigate the efficiency of the organizational learning effect of Municipal Solid Waste (MSW) recycling systems.