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Mathematics and virtual culture: An evolutionary perspective on technology and mathematics education
 Educational Studies in Mathematics
, 1999
"... ABSTRACT. This paper suggests that from a cognitiveevolutionary perspective, computational media are qualitatively different from many of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive evolution suggest that human cogniti ..."
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Cited by 7 (3 self)
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ABSTRACT. This paper suggests that from a cognitiveevolutionary perspective, computational media are qualitatively different from many of the technologies that have promised educational change in the past and failed to deliver. Recent theories of human cognitive evolution suggest that human cognition has evolved through four distinct stages: episodic, mimetic, mythic, and theoretical. This progression was driven by three cognitive advances: the ability to “represent ” events, the development of symbolic reference, and the creation of external symbolic representations. In this paper, we suggest that we are developing a new cognitive culture: a “virtual ” culture dependent on the externalization of symbolic processing. We suggest here that the ability to externalize the manipulation of formal systems changes the very nature of cognitive activity. These changes will have important consequences for mathematics education in coming decades. In particular, we argue that mathematics education in a virtual culture should strive to give students generative fluency to learn varieties of representational systems, provide opportunities to create and modify representational forms, develop skill in making and exploring virtual environments, and emphasize mathematics as a fundamental way of making sense of the world, reserving most exact computation and formal proof for those who will need those specialized skills.
ABSTRACT Title of Dissertation: A COGNITIVE FRAMEWORK FOR ANALYZING AND DESCRIBING INTRODUCTORY STUDENTS ’ USE AND UNDERSTANDING OF MATHEMATICS IN PHYSICS
"... Many introductory, algebrabased physics students perform poorly on mathematical problem solving tasks in physics. There are at least two possible, distinct reasons for this poor performance: (1) students simply lack the mathematical skills needed to solve problems in physics, or (2) students do not ..."
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Many introductory, algebrabased physics students perform poorly on mathematical problem solving tasks in physics. There are at least two possible, distinct reasons for this poor performance: (1) students simply lack the mathematical skills needed to solve problems in physics, or (2) students do not know how to apply the mathematical skills they have to particular problem situations in physics. While many students do lack the requisite mathematical skills, a major finding from this work is that the majority of students possess the requisite mathematical skills, yet fail to use or interpret them in the context of physics. In this thesis I propose a theoretical framework to analyze and describe students’ mathematical thinking in physics. In particular, I attempt to answer two questions. What are the cognitive tools involved in formal mathematical thinking in physics? And, why do students make the kinds of mistakes they do when using mathematics in physics? According to the proposed theoretical framework there are three major theoretical constructs: mathematical resources, which are the knowledge elements that are activated in mathematical thinking and problem solving; epistemic games, which are patterns of
Choosing integration methods when solving differential equations
"... Abstract: There are two common types of solution methods for solving simple integrals: using integration constants or using limits of integration. We use the resources framework to model student solution methods. Preliminary results indicate both problematic and meaningful intersections of physical ..."
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Abstract: There are two common types of solution methods for solving simple integrals: using integration constants or using limits of integration. We use the resources framework to model student solution methods. Preliminary results indicate both problematic and meaningful intersections of physical meaning and mathematical formalism when solving linked integrals. Students solving separable first order differential equations in typically use two methods: an integration constant (“+C”) or limits of integration. The two methods are similar, involving antiderivatives and boundary conditions. But, we have found that students using the integration constant method rarely find a physically complete solution. We use a resources framework (Hammer, 1996, 2000; Sabella & Redish, 2007; Sayre & Wittmann, 2008; Sherin, 1996) to give a finegrain, “knowledgeinpieces ” analysis of student reasoning (diSessa, 1988, 1993). Consider the question asked in Figure 1. Students are given a separable differential equation and a set of unusual boundary conditions. A typical solution of the integrals in time and velocity requires that one carry out indefinite integration on each. In the “+C ” method, the matched equations require a velocity of 366 m/s at 0 s. In the limits method, the time integral runs from t = 0 s to some unspecified time, t, and the velocity integral runs from 366 m/s to some undefined velocity, v. For all the mathematical similarity of the two, our students rarely use limits. We have asked this question (with and without some steps filled in) in several settings, and present examples from three student groups to illustrate problems they have with integration limits and how we might help them. A group is working on the following problem. A bullet fired horizontally has a muzzle velocity of 366 m/s and experiences a –cv 2 air resistance. Find an equation that describes the horizontal velocity of the bullet with respect to time. A student writes: What would you do next?
How to study learning processes? Reflection on methods for finegrain data analysis
"... Abstract: This symposium addresses methodological issues in studying children’s knowledge and learning processes. The class of methods discussed here looks at processes of learning in finegrained detail, through which a theoretical framework evolves rather than is merely applied. This class of meth ..."
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Abstract: This symposium addresses methodological issues in studying children’s knowledge and learning processes. The class of methods discussed here looks at processes of learning in finegrained detail, through which a theoretical framework evolves rather than is merely applied. This class of methodological orientations to studying learning processes diverges from more common ones in several important ways: 1) Attention to diverse features of the learning interaction; 2) conducting a momentbymoment analysis, zooming in on the fine details of the studied processes; 3) rather than proving or applying a theory, the objective is to make theoretical innovations, or to develop a “humble theory. ” The challenge of using such techniques is that, by their nature, they do not follow a strongly delineated procedure, especially not the usual sort of coding. This symposium attempts to begin addressing the methodological issues by reflecting on several cases of data analysis.