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Some Thoughts about
Mathematics 4005: Geometry

Spring 2006

James J. Madden

This a junior-senior level course in geometry. As such, it meets the standards of substance and rigor that apply to any math course at this level. Because this course is required for secondary-teacher certification in mathematics, the audience typically consists mostly of prospective high-school math teachers. Therefore, the course has been designed to help future teachers acquire the understanding of geometry that they will need in their own classrooms. Those intending to teach need a deeper and more extensive understanding of geometry than almost anyone else. Accordingly, this course will contain plenty to interest anyone who wants to learn geometry.

This year, we will use John Stillwell's textbook, The Four Pillars of Geometry, Springer 2005. This course will also make use of historical material. We will read original mathematical sources such as Euclid, Descartes, and others. We will examine carefully how the ideas we are focusing on have changed and developed.

Human knowledge of geometry has been advancing since ancient times, but not through simple accumulation. The growth of knowledge is better described as evolutionary. As in evolution, some forms advance while others die out, and the texture of human understanding changes. History reveals the many faces and characters that the deep and enduring ideas have taken on.

From its earliest beginnings as a subject in its own right, geometry had a unique, somewhat artificial language. We can see this clearly when we read Euclid. The language we use today is separated yet further from common language by over two additional millennia of evolutionary intellectual remodeling. Because of the sheer quantity of thought that is condensed in our modern understanding of geometry, the language has become highly refined and specialized. In vocabulary, in turn of phrase and in pattern of expression and expectation, it is not the language of the dinner table or of the street. Knowing geometry involves being able to use this language.

Learning a subject is not only about learning numerous facts, representations and procedures. It is also about learning how those facts, representations and procedures hang together. With a lot of experience, we gain the ability to navigate a field in an intuitive and automatic way. We also create extensive commentary on what we know and how we know it. Advanced knowledge, therefore, includes not only ready access to relevant content and fluid ability to apply it, but also an at-home-ness in a subject in the sense that we can reflect and talk explicitly about where our knowledge comes from and how it is organized. There is a level of knowing beyond even this. We reach it when we are not only at home in a complex body of knowledge, but are able to recognize the components of this knowledge as they develop in others' minds and can guide others to levels of understanding that equal---or possibly even exceed---our own. One who has reached this level of understanding is called a teacher.

For elaborations of the ideas in the last paragraph, see: Ball, D.L., Lubienski, S. and Mewborn, D. (2001). Research on teaching mathematics: The unsolved problem of teachers' mathematical knowledge. In V. Richardson (Ed.), Handbook of research on teaching (4th ed.). New York: Macmillan. 433-456.

My expectation is that students in this course will develop:

  1. the ability to reason with geometric ideas effectively and precisely, with confidence, flexibility and imagination,
  2. the ability to articulate unified vision of how these ideas hang together with one another, with the rest of mathematics and with the rest human knowledge, and
  3. above all, the ability to explain these ideas confidently, from numerous perspectives, using numerous examples and illustrations and at numerous levels of sophistication and abstraction.

For me to be successful as a teacher, I need to find out what students know, monitor the progress of their learning and respond to theirr needs. The only way for me to get the information I need is to hear from my class. I hope that all class meetings include lively discussion in which all students participate. I encourage every student to speak up, to ask questions and to share ideas.

Despite the importance of modern mathematical language, as discussed above, it is well to remember that much of this this language was crafted to contain or represent what mathematicians know---not necessarily to teach it. If mathematics is a way of knowing, and if mathematicians have been spectacularly effective in recording the remarkable things they have discovered by arranging and displaying them in logical, deductive order using a specialized language, they have been less successful in finding ways to lead ordinary people from the world of natural, largely social, thinking deep into the world of mathematical ideas.

Teaching experience seems to show that the order of logic is seldom the order in which ideas develop naturally in people's minds. Many modern high-school curricula seem to have taken this idea to the limit, pushing the specialized language and the logical and deductive aspects of the discipline into the background and concentrating on making links to direct experience in order to develop intuitions. Critics have pointed out that intuitions alone are not enough. Students who do not know mathematical definitions and who have little notion of the specialized language and of logical structure of geometry simply do not know the subject.

The need to balance intuition and logic in the teaching of mathematics is a classical theme on which Poincaré wrote brilliantly nearly 100 years ago. I do not believe there is an easy solution. Everyone who teaches mathematics faces the challenge of leading people into new world that many find at first unnatural and possibly uncomfortable. To find the best way is not a mathematical problem per se, but a problem in human relationships. This is not to cast it aside. To the contrary, there may be no purely mathematical problem that is as important as this in determining the human value of mathematics.

H. Poincaré. (1905). Intuition and logic in mathematics. In S.J. Gould (ed.) The Value of Science: Essential writing of Henri Poincaré. New York, Modern Library, 2001. 197--209.

H. Poincaré (1908). Mathematical definitions and education. In the previously cited volume. 441--459.