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Mathematics Masterclasses for Young People

Micahel Sewell
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Richard J. Wilders
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Mathematics Masterclasses is based on a series of classes the author facilitated with ten-to-thirteen-year-olds in the United Kingdom. His goal was to introduce interesting mathematics that the children were unlikely to encounter (at least in the near future) in the standard curriculum. The book consists of 89 brief bits of mathematics. Early sections are designed to “provide the reader with an understanding, via a diversity of frequently novel situations, of how mathematics really works and grows in practice.” (p. 2). I think the author provides several examples of how mathematics works, but does not succeed so well at showing how it grows. Sewell, by his own description, did not set out to create a textbook. Instead, these are small peeks into the landscape of mathematics. Each piece is described as a masterclass; I will refer to them as classes.

The examples are eclectic and vary widely in difficulty. Some are as short as a paragraph while others approach three pages. There is not much offered by way of connecting the pieces to one another. In many cases just an outline is provided. I think teachers might make use of these, but am doubtful that young children would profit much from reading them by themselves. The book would be much improved if the author had provided links to additional information on each topic. That way children could browse the book and use the links to explore further according to what interested them. Since many of the word problems are set (naturally enough) in Great Britain, U.S. students may not relate to them as well. As seen below, one of my favorite examples concerned rugby — a sport most U.S. children would be unaware of.

While some of the examples seem to show “how mathematics really works” others, in my opinion, miss the mark. For example, there is a discussion (# 37) of the conversion between Fahrenheit and Centigrade scales and an even longer (#43) discussion of the conversion of the British monetary system to decimal coinage. The temperature scale article has a nice graph which ties the conversion to the idea of a linear function. Similar graphs for the monetary conversion seem to imply (figure 29) that coinage can be infinitely divided. I think these graphs should be discrete, not continuous. Class #46 titled Perimeter-Diameter Ratios is seemingly intended to lead children to Archimedes’ calculation of the value of \(\pi\) using polygons. A short table of values of \(\frac{p}{d}\) is provided for objects described as “thin rectangle”, “thicker rectangle”, and finally circle, with the ratio reported as 3.15. I’m not sure what to make of this even as a starting off point. Why not provide a table of values for the perimeters of inscribed and circumscribed regular polygons?

Among the best is the sequence of master classes 47 through 50 which discuss Fibonacci’s sequence, the golden ratio, and occurrences of the sequence in nature. These are quite well done, but don’t add much to what is available in other texts and online.

There are several nice classes which would challenge all but the brightest children in the book’s target age group. One of my favorites was #68, The Rugby Goal Riddle. Television coverage of rugby includes data like this: distance 41.9 meters, angle 23o, apparent goal width 5.1 meters. The class provides a nice discussion of how to make sense out of these numbers and why the last one is the most interesting. Clearly American football viewers are not deemed intelligent enough to deal with such things as most NFL coverage treats momentum as an adjective describing the team currently doing the best! Class #87, Holditch’s Theorem, is another interesting bit of mathematics but assumes a lot on the part of the reader. An illustration would have helped a lot here.

I enjoyed reading through this set of master classes and found lots to like in it. However, I am not sure it fulfills its purpose. As the examples provided above indicate there is often not enough detail for a child to read this without assistance. I would recommend it as a nice source of starting points for students and teachers. I don’t think it works as a stand-alone guide to mathematics.

To test my conclusions about how the target audience would react to this little book I asked Tommy Kaufman to read it and let me know what he thought. Tommy is 12 and currently working with me on line integrals using Stewart’s calculus text. Tommy gave the book an over-all “grade” of B. He liked some of the examples, but had reactions similar to mine about others. When I offered him the book to keep, he declined. This from a young man who devoured Raymond Smullyan’s To Mock a Mockingbird.

Richard Wilders is Professor of Mathematics and Marie and Bernice Gantzert Professor of the Liberal Arts and Sciences at North Central College. In addition to the standard undergraduate mathematics curriculum, he teaches courses in the history and philosophy of science and works with bright pre-college students interested in mathematics. In his spare time, he enjoys watching college football, dance, and musical theater. 

1. Introduction
2. Spin-up
3. Subject definitions
4. Odds and evens
5. Solving equations
6. Weighing the baby by algebra
7. Prime numbers
8. Don't jump to conclusions
9. Euler's formula
10. Goldbach's guess
11. Perfect numbers
12. Euclid's theorem
13. Mathematical symbols
14. Medicine problem
15. Dramatic dates
16. The foggy day problem
17. Angles
18. Angles inside a triangle
19. Angles inside a quadrilateral
20. Angles inside a polygon with n sides
21. Method for nding the centre of a circle
22. Angles in a sector of a circle
23. Orthocentre
24. Arithmetic progression
25. Leavers from an expanding school
26. Isaac Newton
27. Geometric progression
28. Zeno's paradox
29. Big birthday problem
30. Sundays in February
31. Think of a number
32. Hand shaking
33. Losing money to the bank
34. Supermarket oers: deal, or no deal?
35. Imperial and metric
36. House prices in Maidenhead
37. Fahrenheit and centigrade
38. Small things
39. Leonardslee cake
40. Halving areas
41. Isosceles tiling
42. Roman numerals
43. New money for old
44. Distance measures
45. Weight measures
46. Perimeter-diameter ratios
47. Fibonacci numbers
48. Quadratic equations and the Fibonacci sequence
49. Pascal's triangle and the Fibonacci sequence
50. Golden ratio
51. Pythagoras and the Fleet Air Arm
52. Fermat's Last Theorem
53. Another proof of Pythagoras' Theorem
54. A third proof of Pythagoras' Theorem
55. Another application of Pythagoras' Theorem
56. Pythagorean triples
57. Nautical notation
58. Paper sizes
59. Paper sizes and an innite sequence of triangles
60. Magic squares
61. Binomial squares
62. Some special squares
63. The nine-point circle
64. The thirteen-point circle
65. Cardioid
66. Irregular hexagons and Pappus' theorem
67. Regular hexagons
68. The rugby riddle
69. Family trees in people and bees
70. The tethered goat problem
71. Fencing the bulls
72. Surprises
73. Sewell's spirals
74. Prime diagonals
75. Cubic cusp in the classroom
76. Nature's circles
77. Rainbow
78. Basis and bases of arithmetic
79. Lunes
80. An octet of equal circles
81. Alternative construction of the octet
82. Triangle constructions
83. A mosaic of equal circles
84. Intersection of equal spheres
85. Christmas cracker
86. Ostrich egg
87. Holditch's Theorem
88. A coee shop problem
89. Step waves
90. References
91. Author information