In this paper we consider data from a study in which students shift from linear to quadratic equations in ways that do not conform to established theoretical frameworks. In solving linear equations, the students did not exhibit the ‘didactic cut’ of Filloy and Rojano (1989) or the subtleties arising from conceiving an equation as a balance (Vlassis, 2002). Instead they used ‘procedural embodiments’, shifting terms around with added ‘rules’ to obtain the correct answer (Lima & Tall, 2008). Faced with quadratic equations, the students learn to apply the formula with little success. The interpretation of this data requires earlier theories to be seen within a more comprehensive framework that places them in an evolving context. We use the developing framework of three worlds of mathematics (Tall, 2004, Tall, 2013), based fundamentally on human perceptions and actions and their consequences, at each stage taking into account the experiences that students have ‘met-before’ (Lima and Tall, 2008, McGowen and Tall, 2010). These experiences may be supportive in new contexts, encouraging pleasurable generalization, or problematic, causing confusion and even mathematical anxiety. We consider how this framework explains and predicts the observed data, how it evolves from earlier theories, and how it gives insights that have both theoretical and practical consequences. 相似文献
This paper considers the consensus tacking problem for nonlinear fractional‐order multiagent systems by presenting a PDα‐type iterative learning control update law with initial learning mechanisms. The asymptotical convergence of the proposed distributed learning algorithm is strictly proved by using the properties of fractional calculus. A sufficient condition is derived to guarantee the whole multiagent system achieving an asymptotic output consensus. An illustrative example is given to verify the theoretical results. 相似文献
Motivated by applications to machine learning, we construct a reversible and irreducible Markov chain whose state space is a certain collection of measurable sets of a chosen l.c.h. space . We study the resulting network (connected undirected graph), including transience, Royden and Riesz decompositions, and kernel factorization. We describe a construction for Hilbert spaces of signed measures which comes equipped with a new notion of reproducing kernels and there is a unique solution to a regularized optimization problem involving the approximation of functions by functions of finite energy. The latter has applications to machine learning (for Markov random fields, for example). 相似文献
The use of recorded lecture videos (RLVs) in mathematics instruction continues to advance. Prior research at the post-secondary level has indicated a tendency for RLV use in mathematics to be negatively correlated with academic performance, although it is unclear whether this is because regular users are generally weaker mathematics students or because RLV use is somehow depressing student learning. Through the lens of cognitive engagement, a quasi-experimental pre- and post-test design study was conducted to investigate the latter possibility.
Cognitive engagement was operationalized using the Revised Two-Factor Study Process Questionnaire (R-SPQ-2F), which measures learning approaches on two major scales: surface and deep. In two mathematics courses at two universities, in Australia and the UK, participants were administered the questionnaire near the course start and finish. Overall findings were similar in both contexts: a reduction in live lecture attendance coupled with a dependence on RLVs was associated with an increase in surface approaches to learning.
This study has important implications for future pedagogical development and adds to the sense of urgency regarding research into best practices using RLVs in mathematics. 相似文献
This study brings together the research focused on science education through project-based learning (PBL). This learning project was carried out in a rural learning community and an attempt was made to adapt to the natural resources of the area by organizing educational outings, experimental activities, and encouraging the participation of families. The overall objective is to test the effectiveness of applying the PBL teaching methodology for learning science in a rural learning community. The methodology used has been qualitative, specifically, the participating research has been used and the information has been compiled in a field notebook. The results show that the didactic proposal had good results; showing that, in conclusion, science teaching today should be inclined toward more innovative educational methodologies such as PBL. 相似文献
This work proposes an extended version of the well-known tree-augmented naive Bayes (TAN) classifier where the structure learning step is performed without requiring features to be connected to the class. Based on a modification of Edmonds' algorithm, our structure learning procedure explores a superset of the structures that are considered by TAN, yet achieves global optimality of the learning score function in a very efficient way (quadratic in the number of features, the same complexity as learning TANs). We enhance our procedure with a new score function that only takes into account arcs that are relevant to predict the class, as well as an optimization over the equivalent sample size during learning. These ideas may be useful for structure learning of Bayesian networks in general. A range of experiments shows that we obtain models with better prediction accuracy than naive Bayes and TAN, and comparable to the accuracy of the state-of-the-art classifier averaged one-dependence estimator (AODE). We release our implementation of ETAN so that it can be easily installed and run within Weka. 相似文献
Acidic catecholamine metabolites, which could serve as diagnostic markers for many diseases, demonstrate an importance of accurate sensing. However, they share a highly similar chemical structure, which is a challenge in the design of sensing strategies. A nanopore may be engineered to sense these metabolites in a single molecule manner. To achieve this, a recently developed programmable nano-reactor for stochastic sensing (PNRSS) technique adapted with a phenylboronic acid (PBA) adaptor was applied. Three acidic catecholamine metabolites, including 3,4-dihydroxyphenylacetic acid (DOPAC), 3,4-dihydroxymandelic acid (DHMA) and 3-methoxy-4-hydroxymandetic acid (VMA) were investigated by PNRSS. Specifically, DHMA, which contains an α-hydroxycarboxylate moiety and an adjacent cis-hydroxyl groups on its benzene ring, reports two binding modes simultaneously resolvable by PNRSS. Assisted with the high resolution of PNRSS, direct regulation of these two binding modes by pH can also be observed. A custom machine learning algorithm was also developed to achieve automatic event classification. 相似文献
ABSTRACTA class of semilinear parabolic reaction diffusion equations with multiple time delays is considered. These time delays and corresponding weights are to be optimized such that the associated solution of the delay equation is the best approximation of a desired state function. The differentiability of the mapping is proved that associates the solution of the delay equation to the vector of weights and delays. Based on an adjoint calculus, first-order necessary optimality conditions are derived. Numerical test examples show the applicability of the concept of optimizing time delays. 相似文献