A Mathematician Comes of Age

A Mathematician Comes of Age

Steven G. Krantz
Series: Spectrum
Copyright Date: 2012
Edition: 1
Pages: 156
  • Cite this Item
  • Book Info
    A Mathematician Comes of Age
    Book Description:

    A Mathematician Comes of Age discusses the maturation process for a mathematics student. It describes and analyzes how a student develops from a neophyte who can manipulate simple arithmetic problems to a sophisticated thinker who can understand abstract concepts, can think rigorously, and can analyze and manipulate proofs. Most importantly, mature mathematics students can create proofs and know when the proofs that they have created are correct. Mathematics is distinct from other disciplines in the nature of its intellectual development. The book lays out these differences and discusses their significance.

    eISBN: 978-1-61444-511-1
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-xii)
  3. Preface
    (pp. xiii-xvi)
  4. Acknowledgements
    (pp. xvii-xviii)
  5. CHAPTER 1 Introductory Thoughts
    (pp. 1-26)

    What is mathematical maturity? How can we identify it? Perhaps more importantly, how can we recognize when it is not there, and then determine to do something about it? These are essential questions for any mathematics teacher, and ones that we must learn together to answer.

    Mathematical maturity is elusive: We know what it is, but we do not know how to say what it is. It is important that we find explicit ways to describe it, to discuss it, and to come to terms with it. One of our goals is to recognize students who have the potential for...

  6. CHAPTER 2 Math Concepts
    (pp. 27-40)

    God is in the details. What sorts of problems can be used to ferret out mathematical maturity? What aspects of the mathematics curriculum are essential to mathematical maturity? What activities in the math department are dedicated to the development of mathematical maturity and which are not?

    How can computers play a role in developing mathematical maturity? Are real analysis and abstract algebra and topology and geometry the be all and end-all of mathematical maturity? Are there other aspects of the mathematical pie that can play a productive role here?

    What parts of the basic calculus course can contribute to mathematical...

  7. CHAPTER 3 Teaching Techniques
    (pp. 41-62)

    It is natural to wonder which teaching techniques will contribute to mathematical maturity. Do the tenets of teaching reform have something to show us about the matter? What about OnLine learning?

    These days capstone experiences are in vogue. At the end of a four-year undergraduate education, the capstone experience helps the student draw together material from different courses, and to see connections that were previously unnoticed. It seems likely that such a project could help to push the student along the maturity path.

    There are many new approaches to math teaching, from those of Uri Treisman to those developed at...

  8. CHAPTER 4 Social Issues
    (pp. 63-76)

    As with any human endeavor, the study of mathematics is affected by human issues. We may wonder whether Asperger’s syndrome, or schizophrenia, or manic depression are part and parcel of mathematical talent. Must the developing math teacher become acquainted with these syndromes and learn how to deal with them?

    We also may take an interest in various standardized means of measuring intelligence. The Stanford-Binet IQ test has been widely used to measure intelligence. The SAT and ACT Exams are still used to determine college placement. The Myers-Briggs Index is a standard device for classifying personalities.

    What is the role of...

  9. CHAPTER 5 Cognitive Issues
    (pp. 77-106)

    “Nature versus nurture” is a very old question in the study of child development. Do children become scholastically gifted because of good genes, or because of a supportive environment at home (and at school), or both? Since the math teacher is more a part of the latter aspect than the former, it is natural for teachers to want to understand this dialectic.

    There are many different types of learning: rote learning, learning by trial and error, learning by attacking difficult problems, learning by imitating a master, learning by exploration, learning by experimentation. Which of these is most relevant to developing...

  10. CHAPTER 6 What is a Mathematician?
    (pp. 107-114)

    It seems clear that the role model for a student endeavoring to achieve mathematical maturity is the senior, successful mathematician. Such a person could be an academic professor with a vigorous research program and an international reputation. It could be a successful and innovative worker in the private sector—such as Robert Noyce, inventor of the memory chip and the microprocessor. It could be a leader at one of the many government institutes—such as the National Security Agency (largest employer of math Ph.D.s in the world), or the Institute for Defense Analyses, or Oak Ridge National Laboratory, or Lawrence...

  11. CHAPTER 7 Is Mathematical Maturity for Everyone?
    (pp. 115-118)

    We tend to be a bit self-absorbed. We spend our days in mathematics departments (either at the university or elsewhere) thinking mathematical thoughts. So we are less than fully aware of the world around us.

    As a result, we often do not have a full appreciation of the fact that there are other points of view in the world. Not everyone is passionate about mathematical maturity. How would a butcher or a baker or a literary critic or a chemist view mathematical maturity? Would any of them attach any value to the concept?

    This chapter considers other points of view....

  12. The Tree of Mathematical Maturity
    (pp. 119-120)
  13. Etymology of the Word “Maturity”
    (pp. 121-122)
  14. Bibliography
    (pp. 123-128)
  15. Index
    (pp. 129-136)
  16. About the Author
    (pp. 137-137)