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Purposes and methods of research in mathematics education, Notices, American Mathematical Society Steffe
 In
, 2000
"... Bertrand Russell has defined mathematics as the science in which we never know what we are talking about or whether what we are saying is true. Mathematics has been shown to apply widely in many other scientific fields. Hence, most other scientists do not know what they are talking about or whether ..."
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Bertrand Russell has defined mathematics as the science in which we never know what we are talking about or whether what we are saying is true. Mathematics has been shown to apply widely in many other scientific fields. Hence, most other scientists do not know what they are talking about or whether what they are saying is true. There are no proofs in mathematics education. —Henry Pollak —Joel Cohen, “On the nature of mathematical proofs” The first quotation above is humorous; the second serious. Both, however, serve to highlight some of the major differences between mathematics and mathematics education—differences that must be understood if one is to understand the nature of methods and results in mathematics education. The Cohen quotation does point to some serious aspects of mathematics. In describing various geometries, for example, we start with undefined terms. Then, following the rules of logic, we prove that if certain things are true, other results must follow. On the one hand, the terms are undefined; i.e., “we never know what we are talking about. ” On the other hand, the results are definitive. As Gertrude Stein might have said, a proof is a proof is a proof. Other disciplines work in other ways. Pollak’s statement was not meant as a dismissal of mathematics education, but as a pointer to the fact that the nature of evidence and argument in mathematics education is quite unlike the nature of evidence and argument in mathematics. Indeed, the kinds of questions one can ask (and expect to be able to answer) in educational research are not the kinds of questions that mathematicians might expect. Beyond that, mathematicians and education researchers tend to have different views of the
A Graphic Approach to the Calculus
 Sunburst Inc, USA (for I.B.M. compatible computers
, 1990
"... In recent years, reform calculus has used the computer to show dynamic visual graphics and to offer previously unimaginable power of numeric and symbolic computation. Yet the available technology has far greater potential to allow students (and mathematicians) to make sense of the ideas. A sensible ..."
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In recent years, reform calculus has used the computer to show dynamic visual graphics and to offer previously unimaginable power of numeric and symbolic computation. Yet the available technology has far greater potential to allow students (and mathematicians) to make sense of the ideas. A sensible approach to the calculus builds on the evidence of our human senses and uses these insights as a meaningful basis for various later developments, from practical calculus for applications to theoretical developments in mathematical analysis and even to a logical approach in using infinitesimals. Its major advantage is that it need not be based initially on concepts known to cause student difficulty, but allows fundamental ideas of the calculus to develop naturally from sensible origins, in such a way as to make sense in its own right for general purposes, support the intuitions necessary for applications, provide a meaning for the limit concept to be used later in standard analysis and further, to provide a sensible basis for infinitesimal concepts in nonstandard analysis.
Successful Students’ Conceptions of Mean, Standard Deviation and the Central Limit Theorem. Manuscript submitted for publication
"... This paper is one in an emerging series of studies by members of the “Research in Undergraduate Mathematics Education Community, ” or RUMEC, concerning the nature and development of college students ’ mathematical knowledge. We present analyses of audiotaped clinical interviews with college freshme ..."
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This paper is one in an emerging series of studies by members of the “Research in Undergraduate Mathematics Education Community, ” or RUMEC, concerning the nature and development of college students ’ mathematical knowledge. We present analyses of audiotaped clinical interviews with college freshmen immediately after they completed an elementary statistics course with a grade of
The fundamental theorem of statistics: Classifying student understanding of basic statistical concepts. Unpublished manuscript
, 2007
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A reaction to "A Critique of the Selection of `Mathematical Objects' as Central Metaphor for Advanced Mathematical Thinking" by Confrey and Costa
"... ion, everyday examples, construction of objects, and naming There is a curious statement on page 163. In advocating a "toolbased" approach Confrey and Costa acknowledge that it would share a central feature with the ideas they are criticizing, to wit, "asserting the importance of st ..."
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ion, everyday examples, construction of objects, and naming There is a curious statement on page 163. In advocating a "toolbased" approach Confrey and Costa acknowledge that it would share a central feature with the ideas they are criticizing, to wit, "asserting the importance of structure". But then they say, "However, (in a toolbased approach) rather than demanding departure from activity, the act of seeing similarities in structure across different contexts would be the basis for abstraction." Again, I have said enough about inaccuracies such as the assertion that the people Confrey and Costa are talking about advocate a "departure from activity". What I am concerned about here is the basis for abstraction that Confrey and Costa advocate. I think that focusing on structure is very different from seeing similarities across contexts. Indeed, I can find no other interpretation of the latter phrase than that there are some ideas which have an independent existence, and that they are ...
A Cognitive Analysis of Cauchy’s Conceptions of Continuity, Limit, and Infinitesimal, with implications for teaching the calculus
, 2011
"... Opinions concerning Cauchy’s ideas of continuity, limit, and infinitesimal, and his role in the development of modern analysis are many and varied. Here we complement the range of views with a cognitive analysis of his work based on Merlin Donald’s notion of ‘three levels of consciousness’ and David ..."
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Opinions concerning Cauchy’s ideas of continuity, limit, and infinitesimal, and his role in the development of modern analysis are many and varied. Here we complement the range of views with a cognitive analysis of his work based on Merlin Donald’s notion of ‘three levels of consciousness’ and David Tall’s framework describing the development of mathematical thinking in terms of embodiment, symbolism and formalism. Cauchy lived in an era when modern formal proof from settheoretic axioms did not yet exist. His theoretical framework is based on a blend of geometric embodiment and manipulable symbolism where symbolic processes with sequences of numbers are conceptualized in such a manner that they can be verbalised as infinitesimal concepts. His insights provide the foundations for later developments as axiomatic formulations of both epsilondelta analysis and nonstandard analysis, though in his era there were no formal conceptions of either. When our students are introduced to calculus and analysis, they too build on geometric embodiment and manipulable symbolism. Research has revealed their difficulties with the concepts of continuity, limit and the intuitive notion of infinitesimal. We exploit the framework of superimposed levels of consciousness and the theory of development through embodiment, symbolism, and formalism. We explore the implications of such a framework for our current views on teaching calculus and analysis.
A COGNITIVE ANALYSIS OF CAUCHY’S CONCEPTIONS OF FUNCTION, CONTINUITY, LIMIT, AND INFINITESIMAL, WITH IMPLICATIONS FOR TEACHING THE CALCULUS
"... In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and acti ..."
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In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It reveals the conceptual power of Cauchy’s vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern settheoretic formulation of mathematical analysis. This offers a reevaluation of the relationship between the natural geometry and algebra of elementary calculus that survives in applied mathematics, and the formal set theory of mathematical analysis that develops in pure mathematics and evolves into the logical development of nonstandard analysis using infinitesimal concepts. It counsels us that educational theories developed to evaluate student learning are themselves based on the conceptions of the experts who formulate them. It encourages us to reflect on the principles that we use to analyse the developing mathematical thinking of students, and to make an effort to understand the rationale of differing theoretical viewpoints. 1.
A COMBINED MATHEMATICS LABORATORY AND CLASSROOM ENVIRONMENT
"... a new classroom with laptop computers connected to a university wide server. Students currently have difficulty integrating experiences in classroom and computer laboratory environments. Our laboratory classroom addresses this problem by blending these environments. The laptop computers address soft ..."
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a new classroom with laptop computers connected to a university wide server. Students currently have difficulty integrating experiences in classroom and computer laboratory environments. Our laboratory classroom addresses this problem by blending these environments. The laptop computers address software and networking needs but have a low profile, which helps to maintain a more balanced classroom atmosphere. The objectives of the project are to use the laboratory classroom to implement curricular changes in the department’s major and minor programs, in specific courses which are taken by preservice teachers of mathematics at UWEC and in a newly developed Interdisciplinary Computational Science Program. The principal investigators will achieve the objectives of the project by continuing to adapt the Calculus, Concepts, Computers and Cooperative Learning project. The main DUE themes emphasized in this project are integration of technology in education and teacher preparation. A major outcome of the project is to create an environment conducive to the blending of technology into the teaching and learning of mathematics. Another outcome is that preservice teachers will obtain greater exposure to technology in a setting that exemplifies ways that software and collaboration can be used to teach mathematics.