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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 8 (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.
Understanding coordinated sets of resources: An example from quantum tunneling
 in ‘Proceedings of the Fermi School in Physics Education Research
, 2003
"... In studying student reasoning about quantum physics in the context of tunneling through a barrier, we observe that students commonly use several reasoning resources in conjunction with one another. Our data is gathered in individual student interviews, ungraded quizzes, diagnostic surveys, and exami ..."
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In studying student reasoning about quantum physics in the context of tunneling through a barrier, we observe that students commonly use several reasoning resources in conjunction with one another. Our data is gathered in individual student interviews, ungraded quizzes, diagnostic surveys, and examination questions. We believe that solely a microscopic perspective on the individually used reasoning resources is too narrow to help us understand student reasoning. We also believe that students do not have a coherent, robust (macroscopic) concept of tunneling that can be described as a coordination class. To account for our data, we introduce a mesoscopic description of a coordinated set of resources. We describe a possible coordinated set in quantum tunneling, complete with a readout strategy and net of associated resources and mathematical forms, that a student uses in favor of another possible coordinated set, the resources and forms of which he has available but which he seems not to read out of the given situation.
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.
STUDENTS ' COLLABORATIVE USE OF COMPUTER BASED PROGRAMMING TOOLS IN SCIENCE:
"... Abstract: This paper presents a smallscale study investigating the use of two different computerbased programming environments (CPEs) as modeling tools for collaborative science learning with fifth grade students. We analyze student work and conversations while working with CPEs using Contextual I ..."
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Abstract: This paper presents a smallscale study investigating the use of two different computerbased programming environments (CPEs) as modeling tools for collaborative science learning with fifth grade students. We analyze student work and conversations while working with CPEs using Contextual Inquiry. Findings highlight the differences in activity patterns between groups using different CPEs. Students using Stagecast Creator (SC) did twice as much planning but half as much debugging compared with students using Microworlds (MW). Students working with MW were using written code on the computer screen to communicate their ideas whereas students working with SC were using the programming language to talk about their ideas prior to any programming. We propose three areas for future research. (1) Exploring different types of communication styles as compared with the use of different CPEs. (2) Identifying students' nascent abilities for using CPEs to show functionality in science. (3) Further understanding CPEs ’ design characteristics as to which may promote or hamper learning with models in science.
BRUCE L. SHERIN A COMPARISON OF PROGRAMMING LANGUAGES AND ALGEBRAIC NOTATION AS EXPRESSIVE LANGUAGES FOR PHYSICS
"... ABSTRACT. The purpose of the present work is to consider some of the implications of replacing, for the purposes of physics instruction, algebraic notation with a programming language. What is novel is that, more than previous work, I take seriously the possibility that a programming language can fu ..."
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ABSTRACT. The purpose of the present work is to consider some of the implications of replacing, for the purposes of physics instruction, algebraic notation with a programming language. What is novel is that, more than previous work, I take seriously the possibility that a programming language can function as the principle representational system for physics instruction. This means treating programming as potentially having a similar status and performing a similar function to algebraic notation in physics learning. In order to address the implications of replacing the usual notational system with programming, I begin with two informal conjectures: (1) Programmingbased representations might be easier for students to understand than equationbased representations, and (2) programmingbased representations might privilege a somewhat different “intuitive vocabulary. ” If the second conjecture is correct, it means that the nature of the understanding associated with programmingphysics might be fundamentally different than the understanding associated with algebraphysics. In order to refine and address these conjectures, I introduce a framework based around two theoretical constructs, what I call interpretive devices and symbolic forms. A conclusion of this work is that algebraphysics can be characterized as a physics of balance and equilibrium, and programmingphysics as a physics of processes and causation. More generally, this work provides a theoretical and empirical basis for understanding how the use of particular symbol systems affects students ’ conceptualization.