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Extending the Frontiers of Mathematics: Inquiries into Proof and Argumentation

Key College Publishing
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How do you take an aspiring math major who has performed well in the calculus sequence and introduce him or her to the broad world of mathematics and proofs beyond calculus? In addition, how do you inspire continuing interest in promising students while perhaps gently warning off those strongly disinclined toward the abstract? With Extending the Frontiers of Mathematics: Inquiries into Proof and Argumentation, Ed Burger of Williams College offers an attractive and very plausible approach. The level of sophistication seems just about right for first or second-year college students. The range of topics is thoughtfully selected, and the level of difficulty of the exercises is designed to challenge, not frustrate or discourage.

The author takes as his model a modified Moore method. The core of the book is a series of “prove and extend, or disprove and salvage” exercises. The author aims to provide a consistent structure to the student for approaching mathematical statements, discovering proofs and developing the analytical skills necessary to critique those proofs.

The first chapter addresses puzzles and patterns using ten different puzzle problems as a vehicle for addressing thinking strategies that work to spur insight. The real work begins in the second chapter where, among other things, the author presents as good an introductory discussion of mathematical induction as I have seen anywhere.

The chapter entitled “Infinity” provides an example of how the author proceeds; here he discusses infinite sets and cardinality. Below are three exercises from this chapter that illustrate the scope, and they are all of the “prove and extend, or disprove and salvage” variety:

  1. The set of natural numbers together with zero has the same cardinality as the set of natural numbers.
  2. The set of natural numbers has the same cardinality as the set of rational numbers.
  3. Given a set S, the cardinality of the power set of S is greater than the cardinality of S.

Thrown in for good measure is the question: Is there a set whose cardinality lies strictly between the cardinalities of the natural numbers and the reals? (In all fairness, the author notes in his “Hints, remarks and leading questions” appendix that this is the hardest question in the module, and students should not feel bad about not answering it. (!))

Other topics considered include elementary number theory, geometry, graph theory, probability, and combinatorics. There is a useful appendix called “A Proof Primer” that discusses how to get ideas for a proof, how to put the proof in writing, and how to review it afterwards. Although most students who might use this book would have studied calculus, the text is useful for a broader group because only one or two modules require any knowledge of calculus.

This is a carefully conceived and well-thought-out approach to the “introduction to proof” course, and it’s worth a careful look.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

Date Received: 
Wednesday, April 11, 2007
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Edward B. Burger
Publication Date: 
William J. Satzer

 Introduction: A brief travel guide to the journey ahead.- 1. Puzzles and patterns: A precursor to proofs.-2. Bringing theorems to justice: Exposing the truth through rigorous proof.- 3. Delving into the dependable digits: Counting on counting numbers.- 4. Going around in circles: The art of modular arithmetic.- 5. The irrational side of numbers: A world of nonrepeating digits.- 6. Discovering how to function in mathematics: Moving beyond ordinary relations.- 7. Infinity: Understanding the unending.- 8. Recursively defined functions: The next generation.- 9. Discrete thoughts of counting: Quantifying complicated quantities.- 10. Quantifying uncertainty with probability: A likely story?.- 11. The subtle art of connecting the dots: Edging up to graphs.- 12. Just plane graphs: Drawing without being cross.- 13. Visible and invisible universes: Geometric vignettes.- 14. A synergy between geometry and numbers: Circles and Pythagorean triples.- 15. The mathematical mysteries within a sheet of paper: Unfolding pattern and structure.- 16. Take it to the limit: An initial approach to analysis.- 17. Uninterrupted thoughts of continuity: A jump-free journey.- 18. An abstract world of algebra: Reconciling with your x.- 19. Cycles and curves: Algebraic structure in numbers and geometry.- 20. Further frontiers.-
Appendix 1. Hints, remarks, and leading questions.- Appendix 2. A proof primer.- Appendix 3. Commentary for instructors.- Appendix 4. A math personality questionnaire.

Publish Book: 
Modify Date: 
Wednesday, August 15, 2007