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American Mathematical Monthly - NOVEMBER 1998



Some Characteristics of Eighth Grade Mathematics Classes in the TIMSS Videotape Study
by Alfred B. Manaster
One component of the Third International Mathematics and Science Study (TIMSS) was a videotape study of a representative sample of eighth grade mathematics classes in Germany, Japan, and the United States. This study provided a unique opportunity to examine and contrast the mathematics curriculum delivered in those classes. The author and three of his colleagues analyzed carefully constructed summary descriptions of 90 of these classes. They found aspects of the content of the lessons that provide insight into the potential of the lessons to enhance students' understanding of mathematics. This paper presents a summary of their methods and results. A representative Japanese lesson is analyzed. Special attention is given to the extent of mathematical reasoning in the lessons, the structure and coherence of the lessons, and the nature of the mathematical tasks presented during the lessons. Contrasts among countries and mathematical subjects are reported.

Summary of Conclusions About United States Lessons
  • ·
    • There were no instances of explicit mathematical reasoning in the United States lessons. ·
    • There are more arithmetic lessons in the United States. ·
    • Lessons in the United States are significantly more fragmented. ·
    • There is less use of solver controlled and multi-step problems in the United States lessons.
    For TIMSS generally, visit


    The Newton and Halley Methods for Complex Roots
    by Lily Yau and Adi Ben-Israel
    It is required to solve f(z) = 0 where f is analytic. Let F(x,y):=|f(x+iy)|, and let z_k=x_k+iy_k (the current iterate) be a point where f andf' are nonzero. We give a geometric interpretation of the Newton and Halley iterations for f at z_k.

    Newton: The next iterate is the point closest to z_k on the intersection line of:


    • ·
      • the xy-plane, and ·
      • the plane tangent to the graph of F at (x_k,y_k,F(x_k,y_k)).
      Halley: The next iterate is the zero of the Mobius transformation that osculates the level set of |f| at z_k.


      The Rook on the Half-Chessboard, or How Not to Diagonalize a Matrix
      by Kiran S. Kedlaya and Lenhard L. Ng,
      We study a simple-looking family of matrices whose eigenvalues appear equally innocuous. The diagonalization of these matrices is not quite so innocuous; we present two approaches, one of which requires a four-variable combinatorial identity with a somewhat roundabout proof. We then apply this result to explore two related random walks, summarized as "the rook on the half-chessboard." Given an n-by-n chessboard from which all squares above (but not including) the northwest-southeast diagonal have been removed, a rook moves with probability 1/2 to some square in its row, and with probability 1/2 to some square in its column. We determine the speeds at which this walk, and a related, more efficient walk, become random.

      The Last Round of Betting in Poker
      by Jack Cassidy
      Mathematical analyses of poker usually assume an unrealistic set of circumstances that never occur in real play. Chief among these is that all players have equal hands going into the final draw.

      This paper assumes generalized, unequal hand distributions for two players in the final round of betting. We develop six simple equations for optimal strategies when raising is not allowed, then look at how things change when raising is allowed.

      We apply the equations to a sample hand of Seven Card Stud, and give pointers on how to apply the optimal strategies to real-life poker games.

      Applications of the Universal Subjectivity of the Cantor Set
      by Yoav Benyamini
      Every compact metric space is a continuous image of the Cantor set, i.e., for each compact metric space K there is a continuous map from the Cantor set onto K.

      This classical theorem of Alexandroff and Hausdorff can heuristically be rephrased as saying that "compact sets of data can be continuously encoded by the Cantor set". This principle is a very powerful tool that can be applied to solve a variety of unrelated problems in topology, geometry, and analysis. In this article we present several such applications. The following two examples are typical.

      (i) (R. Grzaslewicz) For each d 1 there is a compact convex set B in R^(d+2) with the property that each compact convex subset of the d-dimensional unit cube is congruent to a face of B.

      (ii) There is a real-valued, bounded, and continuous function f on the real line R with the property that for each doubly infinite sequence (y_n)_(n \ in Z) of real numbers satisfying |y_n |£ 1 for all n, there is a point t \in Rsuch that y_n = f(t + n) for all n \in Z.



      The Group Z_2 x Z_n and Regular Polygonal Paths
      by Apostolos Thoma

      A Quick Cayley-Hamilton
      by Busiso P. Chisala

      The Probability of a Tie in an n-Game Match
      by J. Marshall Ash

      The Converse of the Mean Value Theorem May Fail Generically
      by J. M. Borwein and Xianfu Wang

      A Trio of Triangular Number Theorems
      by John A. Ewell

      Exceptional Objects

      by John Stillwell



      Goodbye, Descartes
      By Keith Devlin.

      Reviewed by Dan Schnabel

      Fourier Analysis and Boundary Value Problems
      By Enrique A. González-Velasco.

      Reviewed by Jeffrey Nunemacher

      Fair Division: From Cake-Cutting to Dispute Resolution
      By Steven J. Brams and Alan D. Taylor.

      Reviewed by William F. Lucas