*José Ferreirós*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691167510
- eISBN:
- 9781400874002
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691167510.003.0005
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter focuses on the ancient Greek tradition of geometrical proof in light of recent studies by Kenneth Manders and others. It advances the view that the borderline of elementary mathematics ...
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This chapter focuses on the ancient Greek tradition of geometrical proof in light of recent studies by Kenneth Manders and others. It advances the view that the borderline of elementary mathematics is strictly linked with the adoption of hypotheses. To this end, the chapter considers Euclidean geometry, which elaborates on both the problems and the proof methods based on diagrams. It argues that Euclidean geometry can be understood as a theoretical, idealized analysis (and further development) of practical geometry; that by way of the idealizations introduced, Euclid's Elements builds on hypotheses that turn them into advanced mathematics; and that the axioms or “postulates” of Book I of the Elements mainly regiment diagrammatic constructions, while the “common notions” are general principles of a theory of quantities. The chapter concludes by discussing how the proposed approach, based on joint consideration of agents and frameworks, can be applied to the case of Greek geometry.Less

This chapter focuses on the ancient Greek tradition of geometrical proof in light of recent studies by Kenneth Manders and others. It advances the view that the borderline of elementary mathematics is strictly linked with the adoption of hypotheses. To this end, the chapter considers Euclidean geometry, which elaborates on both the problems and the proof methods based on diagrams. It argues that Euclidean geometry can be understood as a theoretical, idealized analysis (and further development) of practical geometry; that by way of the idealizations introduced, Euclid's *Elements* builds on hypotheses that turn them into advanced mathematics; and that the axioms or “postulates” of Book I of the *Elements* mainly regiment diagrammatic constructions, while the “common notions” are general principles of a theory of quantities. The chapter concludes by discussing how the proposed approach, based on joint consideration of agents and frameworks, can be applied to the case of Greek geometry.

*Suzanne M. Wilson*

- Published in print:
- 2002
- Published Online:
- October 2013
- ISBN:
- 9780300094329
- eISBN:
- 9780300127539
- Item type:
- chapter

- Publisher:
- Yale University Press
- DOI:
- 10.12987/yale/9780300094329.003.0009
- Subject:
- Sociology, Education

This chapter focuses mostly on the activities inside the classroom. In the late 1980s, for example, Andrew Porter and his colleagues at Michigan State University conducted a large-scale study of ...
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This chapter focuses mostly on the activities inside the classroom. In the late 1980s, for example, Andrew Porter and his colleagues at Michigan State University conducted a large-scale study of mathematics teaching and discovered that elementary mathematics classroom contained four significant features. The first feature they commented upon was that teachers spent most of their time emphasizing skills, noting a lack of “balance.” Students were rarely asked to formulate mathematical problems, hence mathematics received much less instructional attention compared to reading and language. Twenty years later, the Third International Mathematics and Science Study (TIMSS) reported similar findings. The curriculum in the U.S. seemed too watered down and superficial. This chapter focuses on the findings of research carried out on teachers and students and the effectiveness of the teaching methods ascribed to them.Less

This chapter focuses mostly on the activities inside the classroom. In the late 1980s, for example, Andrew Porter and his colleagues at Michigan State University conducted a large-scale study of mathematics teaching and discovered that elementary mathematics classroom contained four significant features. The first feature they commented upon was that teachers spent most of their time emphasizing skills, noting a lack of “balance.” Students were rarely asked to formulate mathematical problems, hence mathematics received much less instructional attention compared to reading and language. Twenty years later, the Third International Mathematics and Science Study (TIMSS) reported similar findings. The curriculum in the U.S. seemed too watered down and superficial. This chapter focuses on the findings of research carried out on teachers and students and the effectiveness of the teaching methods ascribed to them.

*Chris Sangwin*

- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780199660353
- eISBN:
- 9780191748257
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199660353.003.0006
- Subject:
- Mathematics, Educational Mathematics

Computer algebra systems (CAS) automate computation and so are central to automatic assessment. In automating mathematical computation we reveal very interesting issues in elementary mathematics. ...
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Computer algebra systems (CAS) automate computation and so are central to automatic assessment. In automating mathematical computation we reveal very interesting issues in elementary mathematics. This chapter considers the design and use of CAS for computer aided assessment (CAA).Less

Computer algebra systems (CAS) automate computation and so are central to automatic assessment. In automating mathematical computation we reveal very interesting issues in elementary mathematics. This chapter considers the design and use of CAS for computer aided assessment (CAA).

*John A. Adam*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0004
- Subject:
- Mathematics, Applied Mathematics

This chapter focuses on the ray optics of the rainbow. Some of the most powerful tools of mathematical physics were developed explicitly to address the problem of the rainbow and closely related ...
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This chapter focuses on the ray optics of the rainbow. Some of the most powerful tools of mathematical physics were developed explicitly to address the problem of the rainbow and closely related problems. Indeed, the rainbow has served as a benchmark for testing theories of optics. The more successful of those theories now make it possible to describe the rainbow in mathematical terms—that is, to predict the distribution of light in the sky. The chapter first provides an overview of the physical features and historical details relating to the rainbow before discussing the ray theory of the rainbow from an elementary mathematics standpoint. In particular, it considers some relevant numerical values, polarization of the rainbow, and the divergence problem. It also explores related topics in meteorological optics, including the glory, coronas (simplified), and Rayleigh scattering.Less

This chapter focuses on the ray optics of the rainbow. Some of the most powerful tools of mathematical physics were developed explicitly to address the problem of the rainbow and closely related problems. Indeed, the rainbow has served as a benchmark for testing theories of optics. The more successful of those theories now make it possible to describe the rainbow in mathematical terms—that is, to predict the distribution of light in the sky. The chapter first provides an overview of the physical features and historical details relating to the rainbow before discussing the ray theory of the rainbow from an elementary mathematics standpoint. In particular, it considers some relevant numerical values, polarization of the rainbow, and the divergence problem. It also explores related topics in meteorological optics, including the glory, coronas (simplified), and Rayleigh scattering.