Mathematical difficulties as decoupling of expectation and developmental trajectories

Recent years have seen an increase in research articles and reviews exploring mathematical difficulties (MD). Many of these articles have set out to explain the etiology of the problems, the possibility of different subtypes, and potential brain regions that underlie many of the observable behaviors. These articles are very valuable in a research field, which many have noted, falls behind that of reading and language disabilities. Here will provide a perspective on the current understanding of MD from a different angle, by outlining the school curriculum of England and the US and connecting these to the skills needed at different stages of mathematical understanding. We will extend this to explore the cognitive skills which most likely underpin these different stages and whose impairment may thus lead to mathematics difficulties at all stages of mathematics development. To conclude we will briefly explore interventions that are currently available, indicating whether these can be used to aid the different children at different stages of their mathematical development and what their current limitations may be. The principal aim of this review is to establish an explicit connection between the academic discourse, with its research base and concepts, and the developmental trajectory of abstract mathematical skills that is expected (and somewhat dictated) in formal education. This will possibly help to highlight and make sense of the gap between the complexity of the MD range in real life and the state of its academic science

ACER: A Framework on the Use of Mathematics in Upper-division Physics

At the University of Colorado Boulder, as part of our broader efforts to transform middle- and upper-division physics courses, we research students' difficulties with particular concepts, methods, and tools in classical mechanics, electromagnetism, and quantum mechanics. Unsurprisingly, a number of difficulties are related to students' use of mathematical tools (e.g., approximation methods). Previous work has documented a number of challenges that students must overcome to use mathematical tools fluently in introductory physics (e.g., mapping meaning onto mathematical symbols). We have developed a theoretical framework to facilitate connecting students' difficulties to challenges with specific mathematical and physical concepts. In this paper, we motivate the need for this framework and demonstrate its utility for both researchers and course instructors by applying it to frame results from interview data on students' use of Taylor approximations.Comment: 10 pages, 1 figures, 2 tables, accepted to the 2012 PERC Proceeding

Learning, Arts, and the Brain: The Dana Consortium Report on Arts and Cognition

Reports findings from multiple neuroscientific studies on the impact of arts training on the enhancement of other cognitive capacities, such as reading acquisition, sequence learning, geometrical reasoning, and memory

Learning by Seeing by Doing: Arithmetic Word Problems

Learning by doing in pursuit of real-world goals has received much attention from education researchers but has been unevenly supported by mathematics education software at the elementary level, particularly as it involves arithmetic word problems. In this article, we give examples of doing-oriented tools that might promote children\u27s ability to see significant abstract structures in mathematical situations. The reflection necessary for such seeing is motivated by activities and contexts that emphasize affective and social aspects. Natural language, as a representation already familiar to children, is key in these activities, both as a means of mathematical expression and as a link between situations and various abstract representations. These tools support children\u27s ownership of a mathematical problem and its expression; remote sharing of problems and data; software interpretation of children\u27s own word problems; play with dynamically linked representations with attention to children\u27s prior connections; and systematic problem variation based on empirically determined level of difficulty

Teaching mathematics : self-knowledge, pupil knowledge and content knowledge

Mathematical learning is significantly influenced by the quality of mathematics teaching (Hiebert and Grouws 2007). In spite of the evidence for teachers seeking to do what they believe to be in the best interests of their learners (Schuck 2009; Gholami and Husu 2010), research and policy reports (within the UK and beyond) draw attention to insufficient mathematical attainment (Williams 2008; Eurydice 2011). Why is there this discrepancy? On the one hand, teachers are open to improving their professional practices (Escudero and S´anchez 2007), and on the other, the findings of mathematical education research make little or no impact on teachers’ practice (Wiliam 2003), even although teachers themselves think that they are enacting new or revised practices (Speer 2005)

Using Natural Language as Knowledge Representation in an Intelligent Tutoring System

Knowledge used in an intelligent tutoring system to teach students is usually acquired from authors who are experts in the domain. A problem is that they cannot directly add and update knowledge if they don’t learn formal language used in the system. Using natural language to represent knowledge can allow authors to update knowledge easily. This thesis presents a new approach to use unconstrained natural language as knowledge representation for a physics tutoring system so that non-programmers can add knowledge without learning a new knowledge representation. This approach allows domain experts to add not only problem statements, but also background knowledge such as commonsense and domain knowledge including principles in natural language. Rather than translating into a formal language, natural language representation is directly used in inference so that domain experts can understand the internal process, detect knowledge bugs, and revise the knowledgebase easily. In authoring task studies with the new system based on this approach, it was shown that the size of added knowledge was small enough for a domain expert to add, and converged to near zero as more problems were added in one mental model test. After entering the no-new-knowledge state in the test, 5 out of 13 problems (38 percent) were automatically solved by the system without adding new knowledge

Training mathematical skills in school children: Some preliminary results.

The present study is a further development of an earlier research on school failure. The aim is to develop sound, simple, effective evaluation and training procedures for children with difficulties to learn mathematics. A review of research on mathematical knowledge reveals some confusion in using terms and a lack of empirical studies on intervention strategies. Nevertheless, some progress has been done in defining the necessary skills for children to solve mathematical problems. In the current study, two cases of primary school children (ISCED 1) with difficulties to learn mathematics are presented. They were trained to acquire a set of pre-mathematical or mathematical skills. The core of the intervention procedures is Behavior Skills Training (BST), a highly effective technique for teaching individuals with different disabilities a wide variety of skills. Evaluation of the training is carried out by comparing the percentage of attained objectives before (pre-test) and after training (post-test). Future developments of this research are explaine