Let’s Draw a Picture

In an undergraduate analysis course taught by one of the authors, three prompts are regularly given: (i) What do we know? (ii) What do we need to show? (iii) Let’s draw a picture. We focus on the third prompt and its role in helping students develop their confidence in learning how to construct proofs. Specific examples of visual models and their impact on student work are presented.     

Action knowledge and symbolic knowledge. The computer as mediator / Conocimiento basado en la acción y conocimiento simbólico. El equipo informático como mediador

I begin by stating a small set of troublesome but pervasive educational issues. While these have remained in the background during our work, they have served as the impetus for developing and reflecting on possible roles for computer technology in classrooms. In the remaining sections of the paper I propose that the computer can serve as an effective mediator between action and symbol. After setting the scene in the Laboratory for Making Things, I give an extended example of how computer-as-mediator contributes to learning when played out in this rather unusual classroom.     

The effects of morphological semantics on the processing of Japanese two-kanji compound words

The present study investigated the way in which the activation of semantic representations at the morpheme level affects the processing of two-kanji (morpheme) compound words. Three types of Japanese two-kanji compound words were used as stimulus items: (1) words consisting of two kanji representing opposite concepts (e.g., long + short = length), (2) words consisting of two kanji representing similar concepts (e.g., soft + flexible = pliable), and (3) control words consisting of two closely bound kanji (e.g., wild + field = wilderness). Words consisting of kanji of opposite concepts (M = 768 ms for LD and M = 645 ms for naming) were processed more slowly for lexical decision, but not for naming, than words with kanji of similar concepts (M = 743 ms for LD and M = 636 ms for naming), both of which were, furthermore, processed more slowly for lexical decision and naming than the control compound words (M = 716 ms for LD and M = 590 ms for naming). These results were explained in the framework of the multiple-level interactive-activation model as follows. Because kanji morphemes of opposite and similar concepts are semantically activated both as morpheme units and compound word units, semantic representations of the two morphemes and the compound word which they create compete with each other at the concept level, which slows down lexical decision and naming of the compound word.     

Using visual,embodied, and language representations to teach the 5E instructional model of inquiry science

Helping students to use and interpret representations in science is critically important if they are to become scientifically literate and able to engage in the discourses related to understanding scientific issues. The purpose of this case-study is to report on how one Year 5 teacher in a small, city school in Brisbane, Australia used different visual, embodied, and language representations to capture students’ engagement in the inquiry tasks. While the case study showed that the students demonstrated clear understandings of the relationships between the different scientific phenomena they were investigating, there is no evidence that these ways of thinking and talking generalised to other inquiry-science topics.However, the case study does provide insights into how this teacher used different visual, embodied, and language strategies to help the students in his class develop complex understandings of the inquiry-science unit they were studying.     

扭Heisenberg-Virasoro代数的Bosonic表示

主要研究扭Heisenberg-Virasoro代数的Bosonic表示,构造了扭Heisenberg-Virasoro代数的Bosonic表示并研究了其分解.     

算子的外积表示在线性代数教学中的应用

本文首先给出非正交基下线性算子的外积表示,其次通过引入Gram矩阵给出线性算子在非正交基下矩阵表示和外积表示的系数矩阵之间的关系,再次讨论了非正交基下恒等算子的完备性关系,最后给出了几类线性算子的运算。     

Convergence of sequences and series: Interactions between visual reasoning and the learner's beliefs about their own role

This paper examines part of a set of students who were followed during their first-term, first-year studies in formal definition-based real analysis at a British university. It explores the approaches to problems about convergence of sequences and series made by students who have a tendency to include visual imagery in their reasoning. We explore links between the students' mathematical behavior in solving these problems and their perception of their roles as learners. We develop a theory in which the tendency to visualize, coupled with the students' view of their role, can be used to account for their mathematical behavior. This revised version was published online in July 2006 with corrections to the Cover Date.     

列斐伏尔《空间的生产》述评

列斐伏尔的《空间的生产》一书是当代关于空间问题的最为重要的著作之一。他提出一种独特的社会空间理论，用社会和历史来解读空间，又用空间来解读社会和历史，并使用“空间实践——空间的表征——表征的空间”的“回溯式进步”来强调社会——历史——空间三者之间的辩证统一关系。这是对于西方传统认识论的反叛，又是对于马克思辩证法的空间化尝试，也是列斐伏尔自己理论发展一脉相承的产物，将自己的理论指向于“空间革命”——重建一个差异性的空间。     

Intersection of argumentation and the use of multiple representations in the context of socioscientific issues

Using multiple representations and argumentation are two fundamental processes in science. With the advancements of information communication technologies, these two processes are blended more so than ever before. However, little is known about how these two processes interact with each other in student learning. Hence, we conducted a design-based study in order to distill the relationship between these two processes. Specifically, we designed a learning unit on nuclear energy and implemented it with a group of preservice middle school teachers. The participants used a web-based knowledge organization platform that incorporated three representational modes: textual, concept map, and pictorial. The participants organized their knowledge on nuclear energy by searching, sorting, clustering information through the use of these representational modes and argued about the nuclear energy issue. We found that the use of multiple representations and argumentation interacted with each other in a complex way. Based on our findings, we argue that the complexity can be unfolded in two aspects: (a) the use of multiple representations mediates argumentation in different forms and for different purposes; (b) the type of argumentation that leads to refinement of the use of multiple representations is often non-mediated and drawn from personal experience.     

The influence of textbooks on teachers’ knowledge of chemical bonding representations relative to students’ difficulties understanding

Background: Textbooks are integral tools for teachers’ lessons. Several researchers observed that school teachers rely heavily on textbooks as informational sources when planning lessons. Moreover, textbooks are an important resource for developing students’ knowledge as they contain various representations that influence students’ learning. However, several studies report that students have difficulties understanding models in general, and chemical bonding models in particular, and that students’ difficulties understanding chemical bonding are partly due to the way it is taught by teachers and presented in textbooks.

Purpose: This article aims to delineate the influence of textbooks on teachers’ selection and use of representations when teaching chemical bonding models and to show how this might cause students’ difficulties understanding.

Sample: Ten chemistry teachers from seven upper secondary schools located in Central Sweden volunteered to participate in this study.

Design and methods: Data from multiple sources were collected and analysed, including interviews with the 10 upper secondary school teachers, the teachers’ lesson plans, and the contents of the textbooks used by the teachers.

Results: The results revealed strong coherence between how chemical bonding models are presented in textbooks and by teachers, and thus depict that textbooks influence teachers’ selection and use of representations for their lessons. As discussed in the literature review, several of the selected representations were associated with alternative conceptions of, and difficulties understanding, chemical bonding among students.

Conclusions: The study highlights the need for filling the gap between research and teaching practices, focusing particularly on how representations of chemical bonding can lead to students’ difficulties understanding. The gap may be filled by developing teachers’ pedagogical content knowledge regarding chemical bonding and scientific models in general.