Primary Masterclasses in Mathematics by Jack Carr.
Since 1991, with the support of the Royal Society of Edinburgh, a group of teachers and I have run mathematics masterclasses for primary school children (P6 and P7) in the Edinburgh and Glasgow areas. Each series of masterclasses consists of four consecutive Saturdays and four such series are held each year. Typically, about 60 children from several different primary schools attend and the classes are run on a two year cycle so that the children who attend as P6s can participate again the following year. The main aim of the masterclasses is to give primary children the opportunity to use the mathematical skills they have acquired at school in a creative and exciting way and to encourage a positive attitude to mathematics. The topics that we present are radically different from the usual type of material in school curricula. Each class begins with a game. The games all have a mathematical basis, require few skills to play, but require careful analysis to be sure of winning. Examples include:
(a) Magic 15. This is a game for two players. You begin with the numbers 1 to 9. Players take it in turns to pick a number, once a number is taken it cannot be used again. The winner is the first player to hold exactly three numbers which add up to 15. There is a very clever way to play Magic 15, it has something to do with Magic Squares.
(b) Square Go. This is a game for two players played on a square grid. Any size grid will do, a 9 by 9 grid is ideal. The players take turns. The first player puts a cross on any vertex on the grid. The next player puts a different mark on another vertex. The winner is the first player who has four of his own marks which, when joined together, make a square. Squares can be different sizes 22 and orientations and recognizing the existence of a square can be non-trivial.
(c) No Triangles. In this pencil and paper game for two players you begin by drawing out six spots (or nodes) arranged to form the corners of a regular hexagon. Each player uses a different colour of pencil. The players take turns to join any two spots with a straight line (there are 15 lines in all). Once a line is drawn in one colour, the other player cannot use it. The aim of the game is to avoid drawing a triangle of your own colour, between any three of the original six nodes. Remember the loser is the first person to be forced to draw a triangle. Can the game ever be a draw? Why?
After the game comes an investigation. A typical example is the following called the Mathematician's Escape.
You are one of several pupils held in a circular room. Your teacher announces that he has good news and bad news for you. One of you will be given the day off to go out and enjoy the sunshine. The rest will have to stay and help scrub the building clean. You are all to stand in a circle against the wall. Starting at the door and going round clockwise the teacher will then take every second pupil, and she becomes a 'volunteer', puts on some overalls and starts cleaning. That is, he will leave the first pupil and volunteer the second, leave the third and volunteer the fourth and so on going round in the circle, until just one pupil remains. That last pupil will be free to go out and enjoy herself, this lucky pupil is called the survivor. As a mathematician you will need to do a quick head count and then decide the best place to stand. Which position should you stand in, in order to go free?
The problem is posed and time for thought and exploration is given. A strategy is suggested for tackling the problem, but alternative strategies are not rejected, they are encouraged provided they can be justified. The children work through the carefully prepared material, and every so often the class is brought to attention to pool ideas and to think about the results they have found as a group. Important tools such as tabulation and pattern spotting are introduced until the children have solved the problem to their satisfaction. Beyond their entertainment and motivational value, the investigations are an excellent vehicle for developing problems solving strategies and communication of ideas.
Many of the investigations are difficult and parents and teachers have been surprised by the children's ability to solve them. As an example, in the Mathematician's Escape they can identify the escape position if there are 1998 children. I have received many written responses to the classes from children, parents and teachers. It is particularly interesting that a large number of parents and teachers commented on the way in which the children enthused about the mathematical activities to their friends and relatives.
Author: Jack Carr, Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS.