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Teachers' questioning plays an essential role in shaping collective argumentative discourse. This paper demonstrated that rationality dimensions in teacher questions can be assessed by adapting Habermas' three components of rationality. By coordinating Habermas' construct with Toulmin's model for argumentation, this paper investigated how two secondary mathematics teachers used rational questioning to support student participation in collective argumentation. This paper identified various ways in which two participating teachers used rational questioning to support student participation in argumentation via contributions of argument components. The results establish a theoretical connection between the use of rational questions and students' contributions of components of arguments. The results indicated that not all rational questions were associated with a component of argument, and rational questions may additionally support argumentation in general for the development of a culture of rationality. The study has implications in terms of theory and professional development of teachers. [ABSTRACT FROM AUTHOR]

This paper explores how different schools of thought in mathematics education think and speak about preparing mathematics for teaching by introducing and proposing certain metaphors. Among the metaphors under consideration here are the unpacking metaphor, which finds its origin in the Anglo-American school of thought of pedagogical reduction of mathematics; the elementarization metaphor, which has its origin in the German school of thought of didactic reconstruction of mathematics; and the recontextualization metaphor, which originates in the French school of thought of didactic transposition. The metaphorical language used in these schools of thought is based on different theoretical positions, orientations, and images of preparing mathematics for teaching. Although these metaphors are powerful and allow for different ways of thinking and speaking about preparing mathematics for teaching, they suggest that preparing mathematics for teaching is largely a one-sided process in the sense of an adaptation of the knowledge in question. To promote a more holistic understanding, an alternative metaphor is offered: preparing mathematics for teaching as ecological engineering. By using the ecological engineering metaphor, the preparation of mathematics for teaching is presented as a two-sided process that involves both the adaptation of knowledge and the modification of its environment. [ABSTRACT FROM AUTHOR]

Research on learner errors in mathematics education is beginning to focus on how teachers can learn to identify and engage with the reasoning behind these errors. Research on professional learning communities is beginning to show that they present powerful opportunities for on-going teacher collaboration and learning. In this paper, I bring the two areas of research together. Drawing on data from one professional learning community in the Data Informed Practice Improvement Project, I show how teachers in this community came to understand key concepts about learner errors and shifted their ways of talking about learner errors. I identify three important shifts that the teachers made in their learning about learner errors: from identifying to interpreting errors; from interpreting to engaging errors; and from focusing on learner errors to focusing on their own knowledge. I argue that these three shifts suggest a deepening of teachers' thinking in relation to learner errors. [ABSTRACT FROM AUTHOR]

This paper explored (1) the developmental trend of student level of reasoning across grade level in pattern generalization; (2) the mediatory role of task variables in the developmental trend of student level of reasoning within and across tasks; and (3) developmental trend of student level of reasoning associated with strategy use across grade level. A test designed to measure student level of reasoning was given to a sample of 1232 students from grades 4 to 11 from 5 schools in Lebanon. The Structure of the Learned Outcomes or Responses (SOLO) was used as a theoretical model. Results show that student level of reasoning exhibited an increasing trend across clusters of grade levels and that there were several SOLO levels in each cluster of grade levels. Type of task (immediate, near, far) and function type (linear, non-linear) seem to mediate the development of level of reasoning across grade level but the complexity of the task (simple, more complex) did not. Students used several strategies in each cluster of grade levels and the developmental trends of student level of reasoning associated with strategy use were not uniform and varied across clusters of grade levels, thus supporting a neo-Piagetian interpretation of the results. [ABSTRACT FROM AUTHOR]

Mathematics educators have long emphasised the importance of attitudes, beliefs, and feelings towards mathematics, as crucial in motivating (or not) its learning and use, and as influenced in turn by its social images. This paper is about images of mathematics. Our search for advertisements containing such images in UK daily newspapers, during 2006-2008, found that 4.7 % of editions included a 'mathematical' advert, compared with 1.7 % in pilot work for 1994-2003. The incidence varied across type of newspaper, being correlated with class and gender profiles of the readership. Three quarters of advertisements were classified as containing only very simple mathematics. 'Semiotic discursive' analysis of selected advertisements suggests that they draw on mathematics not to inform, but to connote qualities like precision, certainty, and authority. We discuss the discourse on mathematics in advertising as 'quasi-pedagogic' discourse, and argue that its oversimplified forms, being empty of mathematical content, become powerful means for regulating and 'pedagogising' today's global consumers. [ABSTRACT FROM AUTHOR]

In this paper, we explored how engagement developed over time during a proportional reasoning unit for a group of US bilingual Latino/a students, with particular attention to aspects of social and cultural activity that supported students' engagement. Our findings suggest that student mathematical engagement developed primarily as a relational process characterized by students' social relations across time, their understandings about their relationships with mathematics, and the important relations emerging across proportional reasoning ideas. [ABSTRACT FROM AUTHOR]

Mathematical modelling is endorsed as both a means and an end to learning mathematics. Despite its utility and inclusion as a curricular objective, one of many questions remaining about learners' modelling regards how modelers choose relevant situational attributes and express mathematical relationships in terms of them. Research on quantitative reasoning has informed the field on how individuals quantify attributes and conceive of covariational relationships among them. However, this research has not often attended to modelers' mathematization in open modelling tasks, an endeavor that invites further attention to theoretical and methodological details. To this end, we offer a synthesis of existing theories to present a cognitive account of mathematical model construction through a quantity-oriented lens. Second, we use empirical data to illustrate why it is productive for theories of modelling to attend to and account for students' quantitative reasoning while modelling. Finally, we identify remaining challenges to coordinating different theoretical accounts of model construction. [ABSTRACT FROM AUTHOR]

In activities based on proof without words (PWW), we developed, students are given a PWW– a diagram that alludes to the proof of a mathematical theorem. The students work collaboratively to construct a proof alluded by the PWW, and then each student writes and submits a proof attempt. In a 3-year design-based study, we investigate and develop PWW-based activities for advancing upper secondary students' proficiency in constructing proofs. We use the concept of gap-filling as a theoretical framework. In a nutshell, gap-filling is an action of adding information absent in a text that a reader does for sense-making. We inquire whether secondary school students independently construct a proof when a PWW is at their disposal, what characterizes those gaps that students identify and fill, and how PWW design principles influence students' gap-filling. We identified four categories of gaps: key idea, generality, constructional, and figure property justifications. We find that students mainly fill key idea gaps but not generality gaps and that PWWs that presented the construction procedure led more students to fill the constructional gaps. However, PWWs that did not explicitly present some figures' property led more students to fill those justification gaps. We identify five PWW design principles that can enhance secondary students' gap-filling and conclude that a meticulous design paves the way to implement PWW-based activities for fostering mathematical proving. [ABSTRACT FROM AUTHOR]

This study investigated how 46 pre-service teachers (PSTs) planned for differentiation of instruction in mathematics. Content analysis was utilized to explore the differentiation strategies included and student characteristics considered in PST plans and how PSTs used differentiation strategies and student characteristics to differentiate lesson content, process, product, and environment. In addition, a rubric was designed and utilized to analyze the level of detail PSTs provided in their plans. Results indicated that overall, PSTs were developing in their planning for differentiation of instruction, using general terms to describe the modifications they would make to meet student needs. Moreover, results revealed that PSTs included strategies to differentiate lesson content and environment most often in their plans and frequently considered student readiness levels when planning for differentiation. Findings also revealed that PSTs need support in learning how to plan for differentiation based on student cultural backgrounds. Implications for mathematics teacher preparation are discussed. [ABSTRACT FROM AUTHOR]

The purpose of this cross-national study is to understand teachers' views about the meaning of instructional coherence and the ways to achieve instructional coherence. With respect to the meaning of instructional coherence, whereas the majority of U.S. teachers paid attention to connections between teaching activities, lessons, or topics, the majority of Chinese teachers emphasized the interconnected nature of mathematical knowledge beyond the teaching flow. U.S. teachers expressed their views about ways to achieve instructional coherence through managing a complete lesson structure. In contrast, Chinese teachers emphasized pre-design of teaching sequences, transitional language and questioning based on the study of textbooks and students beforehand. Moreover, they emphasized addressing student thinking and dealing with emerging events in order to achieve 'real' coherence. The findings of the study contribute to our understanding about the meaning of instructional coherence and ways to achieve instructional coherence in different cultural contexts. [ABSTRACT FROM AUTHOR]

The present study, carried out in the Nordic countries, examines the characteristics of students' scholastic performance on items containing graphical artefacts, that is, bar graphs, pie charts and line graphs, selected from the Programme for International Student Assessment (PISA) survey test. Graphical analysis of statistical data resulted in the observation of two major categories of performance by the students. The results of cluster analysis also confirmed the two approaches. One approach consists of items perceived as requiring identification, that is, focusing primarily on perceptual elements. The other consisting of items requiring a critical-analytical approach, that is, involving evaluation of the graphical system, active interaction with subject specific operators and forms of expression. The general observation is that the pattern of response is similar for all these countries, with items demanding an identification approach showing comparatively higher scores than for items perceived as demanding a critical-analytical approach. [ABSTRACT FROM AUTHOR]