Procedural and relational understanding of pre-service mathematics teachers regarding spatial perception of angles in pyramids

In this study, we aim to explore the extent of mathematics pre-service teachers’ ability to apply their procedural understanding combined with spatial perception for drawing conceptual conclusions related to angles in a pyramid. The participants are 16 pre-service high school mathematics teachers. They have studied solid geometry during one academic year, solving problems with various 3-D geometric figures including pyramids and engaging in activities designed to develop spatial perception. At the end of the year, they have taken a final test which examines procedural understanding of 3-D geometric figures as well as relational understanding and spatial perception regarding angles in pyramids. The results illustrate that attainments of the majority of the pre-service teachers in problems requiring only procedural understanding are higher than the attainments in problems which require relational understanding. The results also lead to the assumption that relational understanding of learned material requires application of special teaching methods. Hence, we recommend integrating in the syllabus appropriate courses that focus on the development of this type of understanding.     

Fifth graders’ understanding of fractions on the number line

The goal of this research was to examine fifth graders’ understanding of fractions on the number line. This case‐study design focused on the various ways that students represented fractions on number lines. Students responded to task‐based interview questions by identifying fractions as a number on the number line as well as equivalency and problem solving. The tasks were administered individually to 26 fifth‐grade students over a 15‐minute time frame in their respective schools. The two groups of 10‐year‐old students answered most questions in written form with pencil and paper and were often asked to explain how they arrived at an answer. Student performance was highest when instructed to plot ½ on a number line of 0 to 1 as well as naming a fraction less than ½. The students performed lowest when they attempted to plot ½, ¼, and 1 on a number line with a predetermined unit 0 to 1/3. Other low performing concepts consisted of plotting ¼ on a number line from 0‐3, identifying ¼ on a non‐routine number line, and plotting a unit fraction with an equivalent fraction as well as an improper fraction on a common number line.     

Simulation vs. Understanding: A Tension,in Quantum Chemistry and Beyond. Part A. Stage Setting

We begin our tripartite Essay with a triangle of understanding, theory and simulation. Sketching the intimate tie between explanation and teaching, we also point to the emotional impact of understanding. As we trace the development of theory in chemistry, Dirac's characterization of what is known and what is needed for theoretical chemistry comes up, as does the role of prediction, and Thom's phrase “To predict is not to explain.” We give a typology of models, and then describe, no doubt inadequately, machine learning and neural networks. In the second part, we leave philosophy, beginning by describing Roald's being beaten by simulation. This leads us to artificial intelligence (AI), Searle's Chinese room, and Strevens’ account of what a go‐playing program knows. Back to our terrain—we ask “Quantum Chemistry, ? ca. 2020?” Then move to examples of AI affecting social matters, ranging from trivial to scary. We argue that moral decisions are hardly to be left to a computer. At this point, we try to pull the reader up, giving the opposing view of an optimistic, limitless future a voice. But we don't do justice to that view—how could we? We return to questioning the ascetic dimension of scientists, their romance with black boxes. Onward: In the 3rd part of this Essay, we work our way up from pessimism. We trace (another triangle!) the special interests of experimentalists, who want the theory we love, and reliable numbers as well. We detail in our own science instances where theory gave us real joy. Two more examples‐on magnetic coupling in inorganic diradicals, and the way to think about alkali metal halides, show us the way to integrate simulation with theory. Back and forth is how it should be—between painfully‐obtained, intriguing numbers, begging for interpretation, in turn requiring new concepts, new models, new theoretically grounded tools of computation. Through such iterations understanding is formed. As our tripartite Essay ends, we outline a future of consilience, with a role both for fact‐seekers, and searchers for understanding. Chemistry's streak of creation provides in that conjoined future a passage to art and to perceiving, as we argue we must, the sacred in science.     

理解和记忆物理化学公式的方法*

物理化学中涉及的公式多达数百,学生记忆时存在很大困难。为解决这一问题,介绍了几种不同的公式理解和记忆方法,包括演绎推导法、对比记忆法、量纲验证法、理解纠错法、交互验证法,并结合实例,对这几种方法应用做了较为详细的说明。     

The current state of public understanding of nanotechnology

The growing importance of nanotechnology in industry and society has not been accompanied by a widespread understanding of the subject among the general public. Simple questions to initially probe the smallest thing that people can see and can think of reveals a divide in the understanding of the general public. A survey of 1500 individuals ranging in age from 6 to 74 has revealed a lack of knowledge of nanotechnology and especially a lack of understanding of the context of nanotechnology in the world that is too small to see. Survey findings are corroborated by in-depth interviews with 400 adults in studies of nanoscience literacy commisioned by University of California, Berkeley and Cornell in 2002 and 2004, respectively. In general, with the exception of 14–28 year olds, over 60% of respondents say they have never heard of nano or nanotechnology. The results suggest that the general public, especially middle-school children, has no firm foundation to understand nanotechnology and likely will continue to be equally impressed by credible scientific information as well as pure fictional accounts of nanotechnology.     

Conceptual blending: Student reasoning when proving “conditional implies conditional” statements

Conceptual blending describes how humans condense information, combining it in novel ways. The blending process may create global insight or new detailed connections, but it may also result in a loss of information, causing confusion. In this paper, we describe the proof writing process of a group of four students in a university geometry course proving a statement of the form conditional implies conditional, i.e., (p → q) ⇒ (r → s). We use blending theory to provide insight into three diverse questions relevant for proof writing: (1) Where do key ideas for proofs come from?, (2) How do students structure their proofs and combine those structures with their more intuitive ideas?, and (3) How are students reasoning when they fail to keep track of the implication structure of the statements that they are using? We also use blending theory to describe the evolution of the students’ proof writing process through four episodes each described by a primary blend.     

Skewing (n+1) tensor indices where the index range is n

It is known that any totally skew quantity with (n?+?1) indices, each of which ranges over n values, vanishes identically. The aim of this short note is to show that this is equivalent to the simple fact that any (n?+?1) vectors in an n-dimensional vector space are linearly dependent.     

Understanding the integral: Students’ symbolic forms

Researchers are currently investigating how calculus students understand the basic concepts of first-year calculus, including the integral. However, much is still unknown regarding the cognitive resources (i.e., stable cognitive units that can be accessed by an individual) that students hold and draw on when thinking about the integral. This paper presents cognitive resources of the integral that a sample of experienced calculus students drew on while working on pure mathematics and applied physics problems. This research provides evidence that students hold a variety of productive cognitive resources that can be employed in problem solving, though some of the resources prove more productive than others, depending on the context. In particular, conceptualizations of the integral as an addition over many pieces seem especially useful in multivariate and physics contexts.     

Interior of a Schwarzschild Black Hole Revisited

The Schwarzschild solution has played a fundamental conceptual role in general relativity, and beyond, for instance, regarding event horizons, spacetime singularities and aspects of quantum field theory in curved spacetimes. However, one still encounters the existence of misconceptions and a certain ambiguity inherent in the Schwarzschild solution in the literature. By taking into account the point of view of an observer in the interior of the event horizon, one verifies that new conceptual difficulties arise. In this work, besides providing a very brief pedagogical review, we further analyze the interior Schwarzschild black hole solution. Firstly, by deducing the interior metric by considering time-dependent metric coefficients, the interior region is analyzed without the prejudices inherited from the exterior geometry. We also pay close attention to several respective cosmological interpretations, and briefly address some of the difficulties associated to spacetime singularities. Secondly, we deduce the conserved quantities of null and timelike geodesics, and discuss several particular cases in some detail. Thirdly, we examine the Eddington–Finkelstein and Kruskal coordinates directly from the interior solution. In concluding, it is important to emphasize that the interior structure of realistic black holes has not been satisfactorily determined, and is still open to considerable debate.     

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介绍了HL-2M装置控制系统的静态架构、动态行为、控制理论以及模拟仿真。设计采用通讯网络将各控制子系统有机地连接起来,以实现各子系统的实时通讯和控制。等离子体放电控制系统将保证实验的各种参数达到预定要求,监控系统的故障处理机制可以减少各种实验异常对装置造成破坏,磁场控制系统的各种控制器则用来控制等离子体电流和位形,模拟仿真系统可以实现方便地验证控制器的目的。