The 'Problem of Points' and Perseverance

Over the years, the journals of the National Council of Teachers of Mathematics (NCTM) have published numerous articles on the history of mathematics and its use in teaching. These journals include Teaching Children Mathematics, Mathematics Teaching in the Middle School, and Mathematics Teacher. Convergence founding co-editor Frank Swetz has arranged with NCTM to republish in Convergence (in pdf format) two articles from Mathematics Teacher annually. One of the editor’s picks for 2015 is an article by Keith Devlin on the 1654 correspondence between Blaise Pascal and Pierre de Fermat about the “problem of the points”:

Keith Devlin, "The Pascal-Fermat Correspondence: How Mathematics Is Really Done," Mathematics Teacher, Vol. 103, No. 8 (April 2010), pp. 578-582. Reprinted with permission from Mathematics Teacher, © 2010 by the National Council of Teachers of Mathematics. All rights reserved.

The article includes suggestions for how the problem itself can be used to introduce middle school, high school, and college students to probability theory, and how Pascal’s initial confusion about a solution to the problem can be used to motivate students to persevere in mathematics.

Devlin recently recalled how he came to write this article:

The idea for an article in NCTM’s Math Teacher came from someone involved in the magazine’s editorial process — I don’t recall who, so I’ll call him or her X. X had read my then-new book The Unfinished Game, and felt that the story of the Pascal-Fermat correspondence provided an excellent example of the difficulties even the best mathematicians can have grasping a new idea or method. To make it work in the Math Teacher, X said, I would have to provide a lesson plan that a teacher could use in order to make effective use of it in the classroom. I had never written a lesson plan in my life, and it took me a couple of attempts before X was satisfied. “But there is just one more thing,” said X. “Can you provide a version of the paper that prevents the magazine's submission reviewers from identifying the author? They do blind reviewing.” I was willing to comply, though it would clearly not be easy, since their definition of blind reviewing was as tight as could be imagined. In fact, X said, I could not refer to any of my own work. Tricky, since my book was the only readily available, comprehensive, and (I like to think) readable account of that particular episode in the history of mathematics. Still, it was worth the effort to get that great potential educational resource in front of thousands of teachers. I clearly made a good job, since the paper sailed through the reviewing process. In fact, the reviewers' report had just one complaint. They thought it was very remiss of the author not to mention the excellent recent work of Keith Devlin on the subject.

About the Author

Keith Devlin is Executive Director of the Human-Sciences and Technologies Advanced Research (H-STAR) Institute at Stanford University. He is the author of The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter That Made the World Modern (Basic Books, New York, 2008) and many other books.

About NCTM

The National Council of Teachers of Mathematics (NCTM) is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research. It publishes five journals, one for every grade band, as well as one on the latest research and another for teacher educators. With 80,000 members and more than 200 Affiliates, NCTM is the world’s largest organization dedicated to improving mathematics education in prekindergarten through grade 12. For more information on NCTM membership, visit http://www.nctm.org/membership.

Other Mathematics Teacher Articles in Convergence

Patricia R. Allaire and Robert E. Bradley, “Geometric Approaches to Quadratic Equations from Other Times and Places,” Mathematics Teacher, Vol. 94, No. 4 (April 2001), pp. 308–313, 319.

David M. Bressoud, "Historical Reflections on Teaching Trigonometry," Mathematics Teacher, Vol. 104, No. 2 (September 2010), pp. 106–112, plus three supplementary sections, "Hipparchus," "Euclid," and "Ptolemy."

Richard M. Davitt, “The Evolutionary Character of Mathematics,” Mathematics Teacher, Vol. 93, No. 8 (November 2000), pp. 692–694.

Jennifer Horn, Amy Zamierowski, and Rita Barger, “Correspondence from Mathematicians," Mathematics Teacher, Vol. 93, No. 8 (November 2000), pp. 688–691.

Po-Hung Liu, “Do Teachers Need to Incorporate the History of Mathematics in Their Teaching?,” Mathematics Teacher, Vol. 96, No. 6 (September 2003), pp. 416–421.

Seán P. Madden, Jocelyne M. Comstock, and James P. Downing, “Poles, Parking Lots, & Mount Piton: Classroom Activities that Combine Astronomy, History, and Mathematics,” Mathematics Teacher, Vol. 100, No. 2 (September 2006), pp. 94–99.

Peter N. Oliver, “Pierre Varignon and the Parallelogram Theorem,” Mathematics Teacher, Vol. 94, No. 4 (April 2001), pp. 316–319.

Peter N. Oliver, “Consequences of the Varignon Parallelogram Theorem,” Mathematics Teacher, Vol. 94, No. 5 (May 2001), pp. 406–408.

Robert Reys and Barbara Reys, “The High School Mathematics Curriculum—What Can We Learn from History?”, Mathematics Teacher, Vol. 105, No. 1 (August 2011), pp. 9–11.

Rheta N. Rubenstein and Randy K. Schwartz, “Word Histories: Melding Mathematics and Meanings,” Mathematics Teacher, Vol. 93, No. 8 (November 2000), pp. 664–669.

Shai Simonson, “The Mathematics of Levi ben Gershon,” Mathematics Teacher, Vol. 93, No. 8 (November 2000), pp. 659–663.

Frank Swetz, “Seeking Relevance? Try the History of Mathematics,” Mathematics Teacher, Vol. 77, No. 1 (January 1984), pp. 54–62, 47.

Frank Swetz, “The ‘Piling Up of Squares’ in Ancient China,” Mathematics Teacher, Vol. 73, No. 1 (January 1977), pp. 72–79.

Patricia S. Wilson and Jennifer B. Chauvot, “Who? How? What? A Strategy for Using History to Teach Mathematics,” Mathematics Teacher, Vol. 93, No. 8 (November 2000), pp. 642–645.