Tooluse supporting the learning and teaching of the Function concept (original) (raw)
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Context problems in realistic mathematics education: A calculus course as an example
Educational studies in mathematics, 1999
This article discusses the role of context problems, as they are used in the Dutch approach that is known as realistic mathematics education (RME). In RME, context problems are intended for supporting a reinvention process that enables students to come to grips with formal mathematics. This approach is primarily described from an instructional-design perspective. The instructional designer tries to construe a route by which the conventional mathematics can be reinvented. Such a reinvention route will be paved with context problems that offer the students opportunities for progressive mathematizing. Context problems are defined as problems of which the problem situation is experientially real to the student. An RME design for a calculus course is taken as an example, to illustrate that the theory based on the design heuristic using context problems and modeling, which was developed for primary school mathematics, also fits an advanced topic such as calculus. Special attention is given to the RME heuristic that refer to the role models can play in a shift from a model of situated activity to a model for mathematical reasoning. In light of this model-of/model-for shift, it is argued that discrete functions and their graphs play a key role as an intermediary between the context problems that have to be solved and the formal calculus that is developed.
Chapter 8: Is Everyday Mathematics Truly Relevant to Mathematics Education?
Journal for Research in Mathematics Education. Monograph, 2002
Early research in everyday mathematics lent support to diverse and often contradictory interpretations of the roles of schools in mathematics education. As research has progressed, we have begun to get a clearer view of the scope and possible contributions of learning out of school to learning in school. In order to appreciate this view it is necessary to carefully scrutinize concepts of real (as in "real life"), utility (or usefulness), context, as well as the distinction between concrete and abstract.
The role of examples in the learning of mathematics and in everyday thought processes
ZDM, 2011
The purpose of this paper is to present a view of three central conceptual activities in the learning of mathematics: concept formation, conjecture formation and conjecture verification. These activities also take place in everyday thinking, in which the role of examples is crucial. Contrary to mathematics, in everyday thinking examples are, very often, the only tool by which we can form concepts and conjectures, and verify them. Thus, relying on examples in these activities in everyday thought processes becomes immediate and natural. In mathematics, however, we form concepts by means of definitions and verify conjectures by mathematical proofs. Thus, mathematics imposes on students certain ways of thinking, which are counterintuitive and not spontaneous. In other words, mathematical thinking requires a kind of inhibition from the learners. The question is to what extent this goal can be achieved. It is quite clear that some people can achieve it. It is also quite clear that many people cannot achieve it. The crucial question is what percentage of the population is interested in achieving it or, moreover, what percentage of the population really cares about it.
The Role of Contexts in Assessment Problems in Mathematics
2005
... which is based on FreudenthaTs idea of mathematics as a human activity, the primary educational goal ... The students must learn to analyze and organize problem situations and to apply mathematics flexibly in ... to imagine the situ-ation or event so that they can make use of their ...
This case study uses a sociocultural perspective and the concept of appropriation (Newman, Griffin and Cole, 1989; Rogoff, 1990) to describe how a student learned to work with linear functions. The analysis describes in detail the impact that interaction with a tutor had on a learner, how the learner appropriated goals, actions, perspectives, and meanings that are part of mathematical practices, and how the learner was active in transforming several of the goals that she appropriated. The paper describes how a learner appropriated two aspects of mathematical practices that are crucial for working with functions (Breidenbach, Dubinsky, Nichols and Hawks, 1992; Even, 1990; Moschkovich, Schoenfeld and Arcavi, 1993; Schwarz and Yerushalmy, 1992; Sfard, 1992): a perspective treating lines as objects and the action of connecting a line to its corresponding equation in the form y = mx + b. I use examples from the analysis of two tutoring sessions to illustrate how the tutor introduced three tasks (estimating y-intercepts, evaluating slopes, and exploring parameters) that reflect these two aspects of mathematical practices in this domain and describe how the student appropriated goals, actions, meanings, and perspectives for carrying out these tasks. I describe how appropriation functioned in terms of the focus of attention, the meaning for utterances, and the goals for these three tasks. I also examine how the learner did not merely repeat the goals the tutor introduced but actively transformed some of these goals.
The emergence of objects from mathematical practices
Educational Studies in Mathematics, 2013
The nature of mathematical objects, their various types, the way in which they are formed, and how they participate in mathematical activity are all questions of interest for philosophy and mathematics education. Teaching in schools is usually based, implicitly or explicitly, on a descriptive/realist view of mathematics, an approach which is not free from potential conflicts. After analysing why this view is so often taken and pointing out the problems raised by realism in mathematics this paper discusses a number of philosophical alternatives in relation to the nature of mathematical objects. Having briefly described the educational and philosophical problem regarding the origin and nature of these objects we then present the main characteristics of a pragmatic and anthropological semiotic approach to them, one which may serve as the outline of a philosophy of mathematics developed from the point of view of mathematics education. This approach is able to explain from a non-realist position how mathematical objects emerge from mathematical practices.