Students' mental prototypes for functions and graphs

This research study investigates the concept of function developed by students studying English A-level mathematics. It shows that, while students may be able to use functions in their practical mathematics, their grasp of the theoretical nature of the function concept may be tenuous and inconsistent. The hypothesis is that students develop prototypes for the function concept in much the same way as they develop prototypes for concepts in everyday life. The definition of the function concept, though given in the curriculum, is not stressed and proves to be inoperative, with their understanding of the concept reliant on properties of familiar prototype examples: those having regular shaped graphs, such as x2 or sin*, those often encountered (possibly erroneously), such as a circle, those in which y is defined as an explicit formula in x, and so on. Investigations reveal significant misconceptions. For example, threequarters of a sample of students starting a university mathematics course considered that a constant function was not a function in either its graphical or algebraic forms, and threequarters thought that a circle is a function. This reveals a wide gulf between the concepts as perceived to be taught and as actually learned by the students

James J. Kaput (1942–2005) imagineer and futurologist of mathematics education

Jim Kaput lived a full life in mathematics education and we have many reasons to be grateful to him, not only for his vision of the use of technology in mathematics, but also for his fundamental humanity. This paper considers the origins of his ‘big ideas’ as he lived through the most amazing innovations in technology that have changed our lives more in a generation than in many centuries before. His vision continues as is exemplified by the collected papers in this tribute to his life and work

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

What is the object of the encapsulation of a process?

Several theories have been proposed to describe the transition from process to object in mathematical thinking. Yet, what is the nature of this ''object'' produced by the ''encapsulation'' of a process? Here, we outline the development of some of the theories (including Piaget, Dienes, Davis, Greeno, Dubinsky, Sfard, Gray, and Tall) and consider the nature of the mental objects (apparently) produced through encapsulation and their role in the wider development of mathematical thinking. Does the same developmental route occur in geometry as in arithmetic and algebra? Is the same development used in axiomatic mathematics? What is the role played by imagery

The fundamental cycle of concept construction underlying various theoretical frameworks

In this paper, the development of mathematical concepts over time is considered. Particular reference is given to the shifting of attention from step-by-step procedures that are performed in time, to symbolism that can be manipulated as mental entities on paper and in the mind. The development is analysed using different theoretical perspectives, including the SOLO model and various theories of concept construction to reveal a fundamental cycle underlying the building of concepts that features widely in different ways of thinking that occurs throughout mathematical learning

Perkinsus marinus extracellular protease modulates survival of Vibrio vulnificus in eastern oyster (Crassostrea virginica) hemocytes

The in vitro effects of the Perkinsus marinus serine protease on the intracellular survival of Vibrio vulnificus in oyster hemocytes were examined by using a time-course gentamicin internalization assay. Results showed that protease-treated hemocytes were initially slower to internalize V. vulnificus than untreated hemocytes. After 1 h, the elimination of V. vulnificus by treated hemocytes was significantly suppressed compared with hemocytes infected with invasive and noninvasive controls. Our data suggest that the serine protease produced by P. marinas suppresses the vibriocidal activity of oyster hemocytes to effectively eliminate V. vulnificus, potentially leading to conditions favoring higher numbers of vibrios in oyster tissues

Students’ Evolving Meaning About Tangent Line with the Mediation of a Dynamic Geometry Environment and an Instructional Example Space

In this paper I report a lengthy episode from a teaching experiment in which fifteen Year 12 Greek students negotiated their definitions of tangent line to a function graph. The experiment was designed for the purpose of introducing students to the notion of derivative and to the general case of tangent to a function graph. Its design was based on previous research results on students’ perspectives on tangency, especially in their transition from Geometry to Analysis. In this experiment an instructional example space of functions was used in an electronic environment utilising Dynamic Geometry software with Function Grapher tools. Following the Vygotskian approach according to which students’ knowledge develops in specific social and cultural contexts, students’ construction of the meaning of tangent line was observed in the classroom throughout the experiment. The analysis of the classroom data collected during the experiment focused on the evolution of students’ personal meanings about tangent line of function graph in relation to: the electronic environment; the pre-prepared as well as spontaneous examples; students’ engagement in classroom discussion; and, the role of researcher as a teacher. The analysis indicated that the evolution of students’ meanings towards a more sophisticated understanding of tangency was not linear. Also it was interrelated with the evolution of the meaning they had about the inscriptions in the electronic environment; the instructional example space; the classroom discussion; and, the role of the teacher

Who gets the information? Gender, power and equity considerations in the design of climate services for farmers

Central to understanding the usefulness of climate and weather forecasts in support of agricultural decision-making is addressing the issue of who receives what information. Many contend that improved climate forecasts since the late 1990s have had limited impact on smallholder farming communities in Africa and across the developing world. However, power and privilege may determine who has access to appropriate climate and advisory services within those communities. In 2011-2012, we tested this hypothesis in three climate-vulnerable farming communities in the CGIAR Research Program on Climate Change, Agriculture and Food Security semi-arid research site of Kaffrine, Senegal. Therein, we assessed gender-specific vulnerabilities to climate-related shocks, endogenous adaptation strategies, and coping mechanisms. From the gap between vulnerability and local capacity, we deduced farmers’ climate service needs, and then assessed whether these systematically differed between distinct vulnerable sub-groups within the community – chiefly, between male and female farmers. In 2011 we introduced a seasonal climate forecast for the first time in the community, and explored perceptions of forecast access, usefulness and value, by both men and women