Audrey's acquisition of fractions: A case study into the learning of formal mathematics

National standards for teaching mathematics in primary schools in the Netherlands leave little room for formal fractions. However,a newly developed programme in fractions aims at learning formal fractions. The starting point in the development of this curriculum is the students’ acquisition of `numeracy infractions’. In this case study we describe the growth in reasoning ability with fractions of one student in this newly developed programme of 30 lessons during one whole school year. In the study we found indications that the programme and its teaching stimulated the progress of an average performer in mathematics. Moreover we found arguments as to what extent formal operations with fractions suits as an educational goal. This revised version was published online in July 2006 with corrections to the Cover Date.     

Interest-enhancing approaches to mathematics curriculum design: Illustrations and personalization

Two common interest-enhancement approaches in mathematics curriculum design are illustrations and personalization of problems to students’ interests. The objective of these experiments is to test a variety of illustrations and personalization approaches. In the illustrations experiment, students (n?=?265) were randomly assigned to lessons with story problems containing decorative illustrations, contextual illustrations, diagrammatic illustrations, misleading illustrations, or no illustrations (only text [control condition]). Students’ problem-solving performance and attitudes were not affected by illustration condition, but learning was better in the control compared with contextual illustrations. In the personalization experiment, students (n?=?223) were randomly assigned to story problems that were either personalized based on: a survey of their interests, their choice of interest topics, a randomly assigned interest topic, or the original nonpersonalized story problem (control). The findings indicated there were benefits for choice personalization both for performance in the problem set as well as on a later learning assessment.     

Getting at the Mathematics: Sara’s Journal

In this paper, we discuss issues in planning and conducting research into mathematics learning. We emphasize two central themes: (a) the learners mathematics (especially the issues and ideas, in given problem situations, that learners choose to think about and to present) and (b) the kinds of knowledge that learners may be building (including their ideas about what mathematics is, and how people do, learn, use, communicate and understand it). While the first theme is at least partly mathematical, the second interweaves cognition and epistemology. Anchored in student data from an extended classroom teaching experiment in the mathematics of change, we focus on key choices needed to build narratives to help researchers capture, in detail, how students engage with mathematics in extended problem explorations.     

Supplementing floundering text with adjunct displays

Three experiments compared the learning potential of text versus outline and matrix displays. In Experiments 1 and 2, college students read or heard a passage about fish and then studied the text, an outline, or a matrix. In Experiment 3, students heard a passage about wildcats, and then studied text, outline, or matrix displays. In all experiments, the text, outline, and matrix formats were informationally equivalent. However, the two-dimensional matrix appeared more computationally efficient than the linear organized text or outline because it (a) positioned related information about fish or wildcats in closer proximity so that local relations within a single category (such as size) were learned, and (b) organized information spatially so that global relations across categories (such as size and diet) were learned. The learning potential of text, outline, and matrix displays was also examined in combination with variations in thematic organization, amount of study time, and time of testing. The most important and consistent findings were that (a) outline and matrix displays produced greater relational learning than the text, and (b) matrix displays produced greater relational learning than outlines.     

The effect of external representation on constructing computer models of complex phenomena

Computer modeling – creating executable modelsof science domains – has been recognized as animportant teaching method. Still not much isknown about the factors making modelingenvironments effective in use. We investigatethe effect of different externalrepresentations on the construction of computermodels. Representations can significantlyinfluence the processes of modeling. In orderto find the specific benefits of two differentrepresentations, we compare dyads working on acollaborative modeling task using a text-basedmodel representation, in which correctequations are required to make the model run,with others using a graphical representation,in which the model is built by qualitativelylinking variables. The learners, secondaryschool students and modeling novices, workedwith the representations on a task in thedomain of physics. Results indicate that thetwo representations induce differentbehaviours, which are appropriate for differentphases of the modeling process.     

The relationship between phonological awareness and the development of orthographic representations

A training study was conducted to investigatethe relationship between phoneme segmentationability and the development of orthographicrepresentations. Five-year-old children withvarying degrees of phoneme segmentationability were taught to read ten new words byrepeated presentation of the words onflashcards. It was found that those childrenwho were most well equipped to perform phonemesegmentation tasks acquired this new readingvocabulary significantly faster than those whowere less phonemically aware. A series ofpost-tests was implemented to discover thenature of the internal orthographicrepresentations which the children had createdfor the words learned. The results of thesepost-tests demonstrated that the children whowere most phonemically aware had alsointernalised the most detailed orthographicrepresentations, despite needing fewerlearning trials. Salient letters fororthographic storage were predictable from thechildren's phoneme segmentation abilities. This paper provides strong support for thethesis that phonemic awareness is related toorthographic storage as well as alphabeticreading techniques.     

Asymmetric translation between multiple representations in chemistry

Experts are more proficient in manipulating and translating between multiple representations (MRs) of a given concept than novices. Studies have shown that instruction using MR can increase student understanding of MR, and one model for MR instruction in chemistry is the chemistry triplet proposed by Johnstone. Concreteness fading theory suggests that presenting concrete representations before abstract representations can increase the effectiveness of MR instruction; however, little work has been conducted on varying the order of different representations during instruction and the role of concreteness in assessment. In this study, we investigated the application of concreteness fading to MR instruction and assessment in teaching chemistry. In two experiments, undergraduate students in either introductory psychology courses or general chemistry courses were given MR instruction on phase changes using different orders of presentation and MR assessment questions based on the representations in the chemistry triplet. Our findings indicate that the order of presentation based on levels of concreteness in MR chemistry instruction is less important than implementation of comprehensive MR assessments. Even after MR instruction, students display an asymmetric understanding of the chemical phenomenon on the MR assessments. Greater emphasis on MR assessments may be an important component in MR instruction that effectively moves novices toward more expert MR understanding.     

“小+谓词性X”结构的句法表征与表达效用

"小+谓词性X"结构在句法特征上表现为:与动词或实义性名词成分共现;与量词或数量成分共现;与"有点(儿)"共现;形成并列或对举格式。该结构的表达效用体现在话语功能和语用特征两个方面,考察显示"小+谓词性X"结构在话语功能上表现为描写和评价话语信息主体、放大和缩小信息情感意义;在语用特征上表现为新颖性、礼貌性和经济性三个方面。     

Language learners’ representations of Portugal: implications for culture teaching / Las representaciones de Portugal de los estudiantes de portugués: implicaciones en la enseñanza de aspectos culturales

This pedagogically-oriented study investigated cultural representations of Portugal held by 29 learners of Portuguese in a large public university in Malaysia. The main purpose of this study was to demonstrate how these representations could be employed for teaching the cultural component of the Portuguese language programme. During the content analysis, representations of Portugal provided by the respondents were divided into 11 categories. The study then assessed salience and favourability of each category of images. The findings revealed that the language learners’ representations of Portugal were for the most part positive. The images referring to Portuguese people formed the largest and most salient category. Overall, the students’ representations of Portugal tended to be simplistic and included ubiquitous stereotypes. A discussion of pedagogical implications arising from these findings concludes the article.     

The Thinking Frames Approach: a case study of inclusion using student‐generated multiple‐representations

Many teachers struggle to provide equitable opportunities for students with special educational needs (SEN) to learn science concepts in the inclusive classroom. This study examines the experience of teaching in an inclusive classroom using a conceptual change approach, the Thinking Frames Approach (TFA), incorporating the use of discrepant events, social construction of scientific conceptions followed by the production of multiple student‐generated representations of their understanding. An in‐depth case study is presented of the experience of Wayne, a student with complex SEN and the effect on his behaviour and science learning based on video/audio recordings of lessons, teacher journal entries, student artefacts, questionnaire results and interviews. It was found that there were positive effects for Wayne's learning using this approach including improved behaviour, greater feelings of self‐efficacy, increased participation in small group and class discussions and improved outcomes on the same assessment tasks as peers. It is suggested that the structured approach of the TFA, the communication of understanding in different modalities, particularly drawing, and the support of peers enabled Wayne to more deeply engage in construction of understanding and may provide teachers with an easy and effective approach to authentic inclusion where real conceptual gains are made by all students.