Published December 2010,November 2010,February 2011.
All the problems this month have something to do with chance and/or probability. As David Speigelhalter discusses in his article, it's often a problematic topic to teach because it's a tricky topic to learn. In this short article we're going to look at the story of probability as told through the primary curriculum and how this may be enhanced by the sort of activities we offer on NRICH.
The first formal ideas appear in most curricula some way through the second part of the primary school - for example in England in year 5, when children are expected to 'explore doubt and uncertainty and develop an understanding of probability through classroom situations; discuss events using a vocabulary that includes the words equally likely, fair, unfair and uncertain'. But if they haven't experienced some of this informally before they are 9, it's an awful lot to take on board.
As with all concept formation, it's important that learners meet fundamental ideas informally, well in advance of formal teaching. In terms of probabability these fundamental ideas are those of fairness, and the familiarisation of the language of chance. Take a look at the Stage 1 activities - Probable Words, In the Playground, The Car That Passes - which encourage children to share the vocabulary they already know and start to put it in some sort of order of likelihood. These ideas can then build up later into the formal numerical representation of probability, the scale of 0-1, which will be introduced in the lower years of secondary school.
Any game using dice or spinners can be exploited to introduce or reinforce the idea of fairness. Many children (and adults) think that a six is harder to throw than any other number, because in their experience they are required to throw a six before they can start a game. It's quite hard to change their perception but the best chance of doing this is to talk about whether each game is fair and how they know. A good example is Incey Wincey Spider and you might also want to take a look at a non-dice game, the old favourite given a new twist, Scissors, Paper, Rock.
A more sophisticated understanding of what makes a game fair depends on knowing what all the possible outcomes could be. So, for example, if you can win a game by throwing an odd number on a 1-6 dice, is it a fair game? Yes - because there are three possible even numbers but also three possible odd. What about a 2-7 dice? Making a list of what the possibilities are encourages children to work systematically, in this case listing the numbers 2, 3, 4, 5, 6, 7 and categorising them into odd or even. In Domino Pick, Odds or Sixes?, Same or Different?, The Twelve Pointed Star Game and Tricky Track, the idea of creating a list is taken further and requires more sophisticated reasoning because there are two things changing each time. For example, in Domino Pick there are the two numbers to list for each domino. Not so straightforward to make sure you have listed them all.
One way of listing all possible outcomes is by using a tree diagram. This idea is taught formally in the lower years of secondary school, but we can introduce the idea of tree diagrams for sorting much earlier. Take a look at The Hair Colour Game for an engaging introduction. Your children will probably enjoy it ...