First year mathematics undergraduates’ settled images of tangent line

This study concerns 182 first year mathematics undergraduates’ perspectives on the tangent line of function graph in the light of a previous study on Year 12 pupils’ perspectives. The aim was the investigation of tangency images that settle after undergraduates’ distancing from the notion for a few months and after their participation in university admission examination. To this end we related the performances of the undergraduates and the pupils in the same questions of a questionnaire; we classified the undergraduates in distinct groups through Latent Class Analysis; and, we examined this classification according to the Analytical Local, Geometrical Global and Intermediate Local perspectives on tangency we had identified among pupils. The findings suggest that more undergraduates than pupils demonstrated intermediate perspectives on tangency. Also, the undergraduates’ settled images were influenced by persistent images about tangency and their prior experience in the context of preparation for and participation in the examination

Using tasks to explore teacher knowledge in situation-specific contexts

Research often reports an overt discrepancy between theoretically/out-of context expressed teacher beliefs about mathematics and pedagogy and actual practice. In order to explore teacher knowledge in situation-specific contexts we have engaged mathematics teachers with classroom scenarios (Tasks) which: are hypothetical but grounded on learning and teaching issues that previous research and experience have highlighted as seminal; are likely to occur in actual practice; have purpose and utility; and, can be used both in (pre- and in-service) teacher education and research through generating access to teachers’ views and intended practices. The Tasks have the following structure: reflecting upon the learning objectives within a mathematical problem (and solving it); examining a flawed (fictional) student solution; and, describing, in writing, feedback to the student. Here we draw on the written responses to one Task (which involved reflecting on solutions of x+x−1=0 of 53 Greek in-service mathematics teachers in order to demonstrate the range of teacher knowledge (mathematical, didactical and pedagogical) that engagement with these tasks allows us to explore

Teachers’ attempts to address both mathematical challenge and differentiation in whole class discussion

International audienceIn this paper we investigate how a group of six mathematics teachers in Greece deals with the need to balance work on mathematically demanding tasks and differentiation in lesson planning and enactment. Videotaped lessons and pre and post reflection interviews were analysed with a specific focus on whole class discussion. The findings show certain teaching practices that appear to promote both mathematical challenge and differentiation and emerging patterns of actions that make the challenge accessible to students

Las actitudes de los estudiantes hacia dos paradojas famosas. Estrategias emergidas en varios grados escolares

III Congreso Internacional Virtual de Educación Estadística (CIVEEST), 21-24 febrero de 2019. [www.ugr.es/local/fqm126/civeest.html]This study aims at getting insight in the strategies employed by secondary students when they are faced with some famous probabilistic paradoxes. Four different age groups of students participated the study (48 in Grade 8, 63 in Grade 9, 53 in Grade 10 and 49 in Grade 12). All students were given the Bertrand’s Box Paradox and the Gardner’s Two Children Paradox modified in an understandable way for all age groups. The 213 written responses (answers and explanations) were analysed and categorized according to various heuristics, misconceptions and types of strategy that seemed to guide students’ choices. Differences among the various grade levels and also differences between the two problems in students’ responses of the same grade level were explored and discussed.Este estudio tiene como objetivo conocer las estrategias empleadas por los estudiantes de secundaria cuando se enfrentan a algunas paradojas probabilísticas famosas. Participaron en el estudio cuatro grupos de estudiantes de diferentes edades (48-Grado 8, 63-Grado 9, 53-Grado 10, 49-Grado 12). Todos los estudiantes resolvieron la paradoja de Bertrand y la paradoja de los dos niños de Gardner, modificadas de manera comprensible para todos los grupos de edad. Las 213 respuestas escritas (respuestas y explicaciones) se analizaron y categorizaron, de acuerdo con diversas heurísticas, conceptos erróneos y tipos de estrategia que parecían guiar las elecciones de los estudiantes. Las diferencias entre los distintos niveles de grado y también las diferencias entre los dos problemas en las respuestas de los estudiantes del mismo nivel de grado fueron exploradas y discutida

Developing student spatial ability with 3D software applications

This paper reports on the design of a library of software applications for the teaching and learning of spatial geometry and visual thinking. The core objective of these applications is the development of a set of dynamic microworlds, which enables (i) students to construct, observe and manipulate configurations in space, (ii) students to study different solids and relates them to their corresponding nets, and (iii) students to promote their visualization skills through the process of constructing dynamic visual images. During the developmental process of software applications the key elements of spatial ability and visualization (mental images, external representations, processes, and abilities of visualization) are carefully taken into consideration

First year mathematics undergraduates’ settled images of tangent line

This article was published in the serial, The Journal of Mathematical Behavior [© Elsevier]. The definitive version is available at: http://www.sciencedirect.com/science/article/pii/S0732312310000556This study concerns 182 first year mathematics undergraduates’ perspectives on the tangent line of function graph in the light of a previous study on Year 12 pupils’ perspectives. The aim was the investigation of tangency images that settle after undergraduates’ distancing from the notion for a few months and after their participation in university admission examination. To this end we related the performances of the undergraduates and the pupils in the same questions of a questionnaire; we classified the undergraduates in distinct groups through Latent Class Analysis; and, we examined this classification according to the Analytical Local, Geometrical Global and Intermediate Local perspectives on tangency we had identified among pupils. The findings suggest that more undergraduates than pupils demonstrated intermediate perspectives on tangency. Also, the undergraduates’ settled images were influenced by persistent images about tangency and their prior experience in the context of preparation for and participation in the examination

Using tasks to explore teacher knowledge in situation-specific contexts

This article was published in the journal, Journal of Mathematics Teacher Education [© Springer] and the original publication is available at www.springerlink.comResearch often reports an overt discrepancy between theoretically/out-of context expressed teacher beliefs about mathematics and pedagogy and actual practice. In order to explore teacher knowledge in situation-specific contexts we have engaged mathematics teachers with classroom scenarios (Tasks) which: are hypothetical but grounded on learning and teaching issues that previous research and experience have highlighted as seminal; are likely to occur in actual practice; have purpose and utility; and, can be used both in (pre- and in-service) teacher education and research through generating access to teachers’ views and intended practices. The Tasks have the following structure: reflecting upon the learning objectives within a mathematical problem (and solving it); examining a flawed (fictional) student solution; and, describing, in writing, feedback to the student. Here we draw on the written responses to one Task (which involved reflecting on solutions of x+x−1=0 of 53 Greek in-service mathematics teachers in order to demonstrate the range of teacher knowledge (mathematical, didactical and pedagogical) that engagement with these tasks allows us to explore

Stereometry activities with DALEST

This book reports on a project to devise and test a teaching programme in 3D geometry for middle school students based on the needs, knowledge and experiences of a range of countries within the European Union. The main objective of the project was the development (and testing) of a dynamic three-dimensional geometry microworld that enabled the students to construct, observe and manipulate geometrical figures in space and which their teachers used to help their students construct an understanding of stereometr

Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis

The tangent line is a central concept in many mathematics and science courses. In this paper we describe a model of students’ thinking – concept images as well as ability in symbolic manipulation – about the tangent line of a curve as it has developed through students’ experiences in Euclidean Geometry and Analysis courses. Data was collected through a questionnaire administered to 196 Year 12 students. Through Latent Class Analysis, the participants were classified in three hierarchical groups representing the transition from a Geometrical Global perspective on the tangent line to an Analytical Local perspective. In the light of this classification, and through qualitative explanations of the students’ responses, we describe students’ thinking about tangents in terms of seven factors. We confirm the model constituted by these seven factors through Confirmatory Factor Analysis

‘Warrant’ revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation

In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence—namely, in Toulmin’s sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument relates to, proposed a classification of warrants (a priori, empirical, institutional and evaluative). Our approach to analysing teacher arguments proposes an adaptation of Freeman’s classification that distinguishes between: epistemological and pedagogical a priori warrants, professional and personal empirical warrants, epistemological and curricular institutional warrants, and evaluative warrants. Our proposition emerged from analyses conducted in the course of a written response and interview study that engages secondary mathematics teachers with classroom scenarios from the mathematical areas of analysis and algebra. The scenarios are hypothetical, grounded on seminal learning and teaching issues, and likely to occur in actual practice. To illustrate our proposed approach to analysing teacher arguments here, we draw on the data we collected through the use of one such scenario, the Tangent Task. We demonstrate how teacher arguments, not analysed for their mathematical accuracy only, can be reconsidered, arguably more productively, in the light of other teacher considerations and priorities: pedagogical, curricular, professional and personal