Modern “Mathematics Masterclasses” in the United Kingdom originated in 1981 on the initiative of George Porter, the then President of the National Association for Gifted Children. Talent in mathematics, as in music, can be recognised at an early age in children, and it is a public service to encourage it, as is now widely accepted.
In fact the idea really goes back to Michael Faraday, who gave Christmas lectures about science for young people at The Royal Institution of Great Britain in London in 1826. Sir Christopher Zeeman, following upon Porter’s initiative, gave the first series of six one-hour lectures (Mathematics Masterclasses) to young people at The Royal Institution in 1981, about “The Nature of Mathematics and The Mathematics of Nature”.
A consequence has been initiatives, widespread now throughout the United Kingdom, of Mathematics Masterclasses, in particular for age groups from 8 to 18 years of age, and with enthusiastic local organisers. I served for several years on the Committee at The Royal Institution whose role was to encourage those Masterclasses nationally.
A reasonable definition of a Masterclass might be that it is devised for “students” (of whatever age level) who have a ready curiosity about what goes on around them, and an interest in identifying an explanation of what they observe, even if that explanation is not immediately obvious but requires, perhaps, a two- or three-stage process to arrive at a solution. The “speaker” will have an intrinsic interest in drawing out an answer from such students, and also of devising problems from any circumstances that lie within the area just stated. In mathematics, the solution process will normally require the identification of appropriate “variables” to describe the problem, the formulation of suitable relations (equations) between those variables, and then the “solution” of those equations in a way which expresses an unknown quantity entirely in terms of known quantities. That is how mathematics “works”.
Every year in the 1990s in Berkshire, England, sixty 12-year-old pupils were gathered at Mathematics Masterclasses at the University of Reading. Attendees were nominated by their schools and showed an aptitude for maths. Two parallel sessions were held, each containing 30 pupils, a lecturer, and qualified helpers.
A typical Masterclass might last for up to three hours (with refreshment breaks, and tutorial sessions, interspersing the lecture material) and broken up into three sessions. Ideally there will be several volunteer teachers circulating to give advice during the tutorial sessions. Teachers from the participating schools were readily found to be enthusiastic to volunteer for this role.
Examples of topics treated in Masterclasses have been “Weather” (the atmosphere and forces therein) by Sir Brian Hoskins, “Water Waves” (in deep and shallow water, and in groups) by Winifred Wood, and the “Dynamics of Dinosaurs” (e.g. their weight and speed) by Michael Sewell. I also gave a Masterclass about “Balloons and Bubbles”, which used mathematics allied to classroom demonstrations to illustrate an associated sequence of topics: pressure, equilibrium of a spherical bubble, tension in a soap film, tension in rubber, pressure peaks and pits, and cylindrical balloons.
The long-term benefit of a Masterclass, and one of its objectives, is to encourage a lasting enthusiasm and curiosity about how to devise a “model” of a natural phenomenon by using mathematics, and thereby to develop the capacity for original thinking about an observed situation in nature, and which is still within the scope of schoolchildren.
An example of an everyday problem suitable for a Masterclass is the following “Coffee Shop Problem”, actually posed to me by my wife in that situation. Given eight points equally distributed around a circle, how many differently shaped triangles can be drawn using only three of those points as vertices? Now generalise the problem by introducing more equally spaced points, and looking for different polygons (not just triangles). This teaches one how to realise that any given problem may be the start of a much larger problem, which is an important part of any mathematical investigation, and which may not be at first apparent.
A further example of a Masterclass problem is the following. Draw a right-angled triangle with unequal shorter sides. Draw three circles, each using one of those sides as the diameter. The two external regions between the larger circle and (in turn) the two smaller circles are called lunes (because they each have the shape of a crescent Moon). Now, prove Hippocrates Theorem (c. 410 B.C.), that the sum of the areas of external lunes is equal to the area of the right-angled triangle.
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