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About Mathematics 4005: Geometry

Spring 2004

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. For many years, the audience has consisted mostly of prospective high-school math teachers and therefore the course has been designed to help such people acquire the understanding of geometry that they will need in their own classrooms. Nonetheless, this course should be attractive to a broader audience, for it is a fact that those intending to teach need a deeper and more extensive understanding of geometry than almost anyone else. A course like this that aims to support future teachers, accordingly, will contain plenty for all who want to learn about geometry.

There are some ideas that are so profoundly intertwined with the roots of geometric reasoning that they exert their influence almost everywhere. They can be observed in the geometry presented to school children, in the thinking of the great masters of the discipline and as driving forces in current research. Like the great themes of literature, they are universal. The backbone of the course is a selection of such important geometric ideas. Listing them by key phrases together with some associated names, the ideas that I have chosen to feature in this course are:

  1. Geometry as an organized body of knowledge (Euclid, Hilbert, Russell, etc)
  2. Ratio, similarity, rational and real numbers (Eudoxus, Euclid, Bolzano, Dedekind, etc.)
  3. Coordinates and algebraic geometry (Descartes, Zariski, Weil, etc.)
  4. Transformations and symmetry (Jordan, Klein, Poincaré, etc.)
  5. The local and the global (Newton, Riemann, etc.)

In drawing up this list, I was influenced by an overview [1] of geometry presented by Shiing-Shen Chern, one of the great geometers of our times. Over the years, I have piloted numerous versions of this course. Experience has led me to modify Chern's list slightly to suit the specific needs of the students I tend to encounter. I do not claim that my list is complete or final in any sense, but it is adequate to provide a conceptual compass. As to whether these ideas are as truly fundamental as I claim, the course itself will provide the evidence. Each of the five big ideas will be broken down into a number of topics. The finer structure of the course is outlined in the following section.

[1] S.-S. Chern, What is geometry?, American Mathematical Monthly 97 (1990), 679-686.


  1. Geometry as an organized body of knowledge
    • Ideas associated with the area formula for a triangle
    • The theory of quadrilaterals
    • Euclid, Book I
    • Modern axiomatizations, formal logic and model theory
  2. Similarity, ratio, real numbers
    • Sea Island theorems
    • Euclid on similarity and ratio
    • The geometry of the real number line
  3. Coordinates
    • What is a coordinate system?
    • Descartes' Geometry
    • Conic sections
  4. Symmetry and transformations
    • Isometries: geometric ideas
    • Transformations and coordinates: complex numbers, matrices
    • Groups of transformations
  5. Local/global
    • Newton's Principia and the calculus
    • Measurement, accuracy and limits
    • Other topic TBA


As an entry-way to each of the topics, I plan to use materials drawn from high-school geometry curricula. I want to get to the big ideas by getting you to think about how to explain basic notions and the logic, images, relations and procedures that they involve. For example, our first problem will be, ``How would you explain the formula for the area of a triangle: area is one-half base time height?'' This leads immediately to further questions, such as: ``What are some alternatives to your explanation? What basic ideas are involved? What must a teacher know in order to make full use of the pedagogical opportunities in contemporary texts? What is the logical structure of the concept of area? How do the ideas involved in this formula generalize? What is the place of the concept of area in geometry?''

This course will make extensive use of historical material. We will read original mathematical sources such as Euclid, Descartes, Riemann and Poincaré, and 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 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. One of the textbooks I have asked you to purchase presents many parts of geometry using the modern language. It is my expectation that you will become comfortable using 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: [2] 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.

Learning goals and grading

My expectation is that students in this course will develop:

  1. the ability to reason with these 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 you know, monitor the progress of your learning and respond to your needs. The only way for me to get the information I need is to hear from you. 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. However, such informal monitoring is not enough. I plan to collect extensive written information from you. In particular, every topic will be accompanied by a pre- and a post-test and often writing opportunities in between.

Student work will be displayed at the course web site. I will not include names, so it will be unlikely that casual visitors to the course web site will be able to identify authors. However, there is no guarantee of anonymity, so if you want to ensue that something is not displayed you should note this on your work. Unless you explicitly ask that a piece of work should be kept private, I will assume permission to display your work (with your name removed, of course).

Grading in this course will be based on post-tests (1/3), assignments (1/3) and midterm and final exams (1/3). As far as possible, I will design tests, assignments and exams to provide evidence of the three abilities I listed above. I will use a 90%-80%-70% scale to assign final letter grades based on final weighted average percents.

Additional thoughts on language, logic and intuition

Despite the importance of modern mathematical language (see above), it is well to remember that much of this this language was crafted to contain or represent what we 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, [3], [4]. 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.

[3] 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.

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

Additional comments on pedagogical style

Dick Stanley, designer and director of professional development programs for high school mathematics teachers offered through the University of California, Berkeley and the University of Texas Austin, has proposed a strategy for studying mathematics that he calls extended problem analysis. In this method, a person (or a class) begins with a specific problem of the kind that occurs routinely in high school mathematics. One solves it, then one seeks out and collects alternate approaches or methods. The alternatives are examined and compared, and relations among the approaches are brought to light. After this, one varies the problem or poses related problems. One examines how the solution methods change, while watching for the underlying ideas and seeking yet further approaches. After numerous variants of the problem have been examined, generalizations begin to present themselves---perhaps a variable can be used in place of a constant, or perhaps some of the conditions can be relaxed. Attention moves to the generalizations and to how the solution methods may be extended. The cycle repeats, and yet higher generalizations are formulated and solved.

In general, the philosophy is to beat the problem to death. But one does not stop there. One continues beating till the problem stands back up, sprouts wings and takes off. In extended problem analysis, one does not look at a mathematical problem as something to be dispatched as rapidly as possible. Instead, each problem is an opportunity to grow ideas. One is not a hunter looking for a quick kill. One becomes a farmer who tends the land, and reaps harvest after harvest.

In this course, we are going to use a modification of Dick Stanley's approach. We will not always begin with the kind of problem that might be assigned to a high-school student. We'll include pedagogical problems, too. They might be of the form, ``How would you teach such and such an idea?'' Or they might be problems that force us to reflect deeply about ideas that you have taken for granted or used without pausing for deep reflection: ``What is the plane of Euclidean geometry?'' ``What is a coordinate system?'' ``What is a real number?'' Answering questions like these force us to do more than find a strategy. They require a certain amount of philosophical reflection. Some of you may be coming to this course with the attitude that there is a certain circumscribed piece of knowledge that, as a teacher of geometry, it will be your task to master and to convey to your future students. You might even feel that the so-called ``standards movement'' in education supports this view. This would be a serious misunderstanding of what standards are intended to do. Let me make it clear that I do not see anything wrong with setting standards, and certainly where standards have been accepted and set, they ought to be met. What is wrong is to view standards as limits, or to believe that having covered the standards we have learned all that needs to be learned or taught all that needs to be taught and can stop. Perhaps it's not so easy to see the point of going so deeply into things as Dick Stanley's approach (or our modification of it) forces us. I would respond that my experiences with high-school students show that many are capable of thinking with exquisite logical detail and precision and that many care about very fine points as well as deep philosophical principles. A teacher should be able to respond to these concerns. A teacher's competence needs to be as broad as the potential interests and concerns of the students.