Students’ reported justifications for their representational choices in linear function problems: an interview study

Thirty-six secondary school students aged 14–16 were interviewed while they chose between a table, a graph or a formula to solve three linear function problems. The justifications for their choices were classified as (1) task-related if they explicitly mentioned the to-be-solved problem, (2) subject-related if students mentioned their own characteristics as representational users, (3) context-related if contextual features surrounding the choice were mentioned and (4) representation-related if formal characteristics of the representations were pointed out. Justifications were mostly task- and subject-related, although contextual and representational features also played an important role. Some students (reportedly) tried to reconcile different (task-, subject-, context- and representation-related) factors before selecting a representation, which was interpreted as an attempt to use their meta-representational competence to make appropriate representational choices. The influence of the didactical contract and the experimental contract on students’ representational choices, as well as the tensions between them, are also discussed.     

“中国美国新加坡”小学数学教材中的“分数定义”

选取中国、新加坡、美国的小学数学主流教材,比较教材中分数定义的相关内容。发现3个国家教材中出现的定义种类大体相同,都包含4种：份数定义、商的定义、数线定义、比的定义,但定义的表征形式及呈现顺序有很大不同,对中国的教材编写及教学设计有一定的借鉴意义。     

Assisting self-explanation prompts are more effective than open prompts when learning with multiple representations

Learning with multiple representations is usually employed in order to foster understanding. However, it also imposes high demands on the learners and often does not lead to the expected results, especially because the learners do not integrate the different representations. Thus, it is necessary to support the learners’ self-explanation activity, which concerns the integration and understanding of multiple representations. In the present experiment, we employed multi-representational worked-out examples and tested the effects of two types of self-explanation prompts as help procedures for integrating and understanding multiple representations. The participants (N = 62) learned about probability theory under three conditions: (a) open self-explanation prompts, (b) self-explanation prompts in an assistance-giving-assistance-withholding procedure (assisting self-explanation prompts), or (c) no prompts (control group). Both types of self-explanation prompts fostered procedural knowledge. This effect was mediated by self-explanations directed to domain principles. Conceptual knowledge was particularly fostered by assisting self-explanation prompts which was mediated by self-explanations on the rationale of a principle. Thus, for enhancing high-quality self-explanations and both procedural knowledge and conceptual understanding, we conclude that assisting self-explanation prompts should be provided. We call this the assisting self-explanation prompt effect which refers to the elicitation of high-quality self-explanations and the acquisition of deep understanding.     

The structure of students’ beliefs about the use of representations and their performance on the learning of fractions

Cognitive development of any concept is related with affective development. The present study investigates students’ beliefs about the use of different types of representation in understanding the concept of fractions and their self‐efficacy beliefs about their ability to transfer information between different types of representation, in relation to their performance on understanding the concept. Data were collected from 1701 students in Grade Five to Grade Eight. Results revealed that multiple‐representation flexibility, ability on solving problems with various modes of representation, beliefs about the use of representations and self‐efficacy beliefs about using them constructed an integrated model with strong interrelations in the different educational levels. Confirmatory factor analysis affirmed the existence of differential effects of multiple‐representation flexibility and problem‐solving ability in respect to cognitive performance and the existence of general beliefs and self‐efficacy beliefs about the use and the role of representations. Results suggested the invariance of this structure across primary (Grades Five and Six) and secondary education (Grades Seven and Eight). However, there are interesting differences concerning the interrelations among those cognitive and affective factors between primary and secondary education.     

Does multimedia learning theory extend to middle-school students?

The purpose of this study is to translate principles of multimedia learning from college-age readers to middle grade students, when reading science texts with a supporting diagram. In this experimental study, sixth-grade students (n = 180) were randomly assigned to display conditions before reading. Each student read two explanatory sciences passages, a life-science and a physical science text. Passages were accompanied by either no illustrations (control), illustrations of the cycle with labels for each part (parts), illustrations of the cycle with labels for each major process (steps), or illustrations showing the labels for each part and each major process (parts and steps). Additionally, there were two text conditions in which half of the students read standard text (control) and half read texts with cues which indicated to students when to access the diagrams (cued). Through ANOVA analysis, in the life-science text students showed modest improvement (partial η² = .18) from the addition of diagrams, with the parts diagram and the steps diagram outperforming the control. In the physical science text, students did not receive benefit from the diagrams. Findings did not replicate results from college-age readers to younger readers, nor between the two texts with younger readers. These results raise concern for the application of multimedia design theory to classroom practice.     

Encrypted objects and decryption processes: problem-solving with functions in a learning environment based on cryptography

This paper introduces an applied problem-solving task, set in the context of cryptography and embedded in a network of computer-based tools. This designed learning environment engaged students in a series of collaborative problem-solving activities intended to introduce the topic of functions through a set of linked representations. In a classroom-based study, students were asked to imagine themselves as cryptanalysts, and to collaborate with the other members of their small group on a series of increasingly difficult problem-solving tasks over several sessions. These tasks involved decrypting text messages that had been encrypted using polynomial functions as substitution ciphers. Drawing on the distinction between viewing functions as processes and as objects, the paper presents a detailed analysis of two groups’ developing fluency with regard to these tasks, and of the aspects of the function concept underlying their problem-solving approaches. Results of this study indicated that different levels of expertise with regard to the task environment reflected and required different aspects of functions, and thus represented distinct opportunities to engage those different aspects of the function concept.     

Shift-invariance of short-time Fourier transform in fractional Fourier domains

Fractional Fourier domains form a continuum of domains making arbitrary angles with the time or frequency domains on the time-frequency plane. Signal representations in these domains are related to the fractional Fourier transform (FrFT). In this paper, a new proof on the shift-invariance of linear time-frequency distributions on fractional Fourier domains is given. We show that short-time Fourier transform (STFT) is the unique linear distribution satisfying magnitude-wise shift-invariance in the fractional Fourier domains. The magnitude-wise shift-invariance property in arbitrary fractional Fourier domains distinguishes STFT among all linear time-frequency distributions and simplifies the interpretation of the resultant distribution as shown by numerical examples.     

空间生产理论视域下的文庙教育空间实践

文庙是中国古代重要的官方教育机构。本文以江阴文庙为例,运用法国哲学家列斐伏尔的社会空间生产理论,对古代中国教育空间的生产进行分析。研究发现:文庙作为一个教育空间,其物理空间的生产实践是在国家制度、权力意志、经济资本和地理环境等多重因素相互作用下的结果;通过相关建筑物的命名与空间环境的营造,文庙空间有效地表征了儒家的教育理想与追求;并且通过官员谒庙、授业课试和定期祭祀等教化活动,文庙所表征的教育空间也得到了儒家知识分子与社会的广泛认同。以史为鉴,江阴文庙空间的生产也为现代教育空间的改造提供了一定的借鉴与启示。     

The Positioning of Indigenous People in Australian History: A Historiography of the 1868 Aboriginal Cricket Tour of England

Abstract

The first international tour by an Australian sports team was the 1868 tour of England undertaken by a group of Aboriginal cricketers. In the following 148 years there have been many histories of the tour written. This paper undertakes a historiographical examination of the tour, contextualizing the writings and their place in Australian society. In doing so, it uncovers the phases of reporting, forgetting, rediscovery, exploiting, ambivalent times, and self-determination that representations of the tour go through. It is argued that these phases provide an analogy for the positioning of Indigenous people and Indigenous history in Australian society.     

Identities, social representations and critical thinking

This comment on L. Simonneaux and J. Simonneaux paper focuses on the role of identities in dealing with socio-scientific issues. We argue that there are two types of identities (social representations) influencing the students’ positions: On the one hand their social representations of the bears’ and wolves’ identities as belonging to particular countries (Slovenia versus France for bears, France and Italy for wolves), in other words, as having national identities; on the other hand representations of their own identities as belonging to the field of agricultural practitioners, and so sharing this socio-professional identity with shepherds and breeders, as opposed to ecologists. We discuss how these representations of identities influenced students’ reasoning and argumentation, blocking in some cases the evaluation of evidence. Implications for developing critical thinking and for dealing with SSI in the classrooms are outlined.