19 August 2016
Down on their luck: are we too quick to discard dice in the classroom?
Are you feeling lucky today?
Fancy trying your luck? Name a number between 1 and 6 inclusive. Now virtually roll a six-sided die (known as a D6 in the business) and see if you were right.
Did you call it?
Want to try again? Probably not. This isn't that interesting a game, and you're more likely to be wrong than not. We've used six-sided cube-shaped dice in mathematics lessons to simulate probability so much you've probably just stifled a yawn (or not read this far). But there's more to dice than just neat tables of sixths and thirty-sixths, and they've been around for longer than you might think.
When we picture a die or a dice (language evolves and that's ok, people), most of us conjure an image of a standard red-and-white cube with pips arranged thus:
But did you know why? And what other kinds of dice there are out there?
You may already know that the earliest types of dice were the hoof bones of animals such as sheep, which had four curved 'sides' that could come to rest on a flat surface . The sides were generally not marked as they all looked different. These dice were called astrolagi (singular astrolagus) which means ankle bone; some etymologists suggest a connection to a traditional cube-shaped die by tracing this through the Arabic translation kab to Greek kubos and English cube. The arrangement of 'pips' (the dots on a die) we use today was first used in around 1300BC - many, many years before Arabic numerals were invented. This was the time of Tutankhamen and Moses; before Pythagoras and before zero. That picture of in your head of five pips arranged symmetrically is older than the numeral '5', and we can only assume that the symmetry is helpful in 'subitising', or counting the dots without thinking about it.
Dice were seen in several cultures as simultaneously embodying chance and fate; as Julius Caesar is reported to have said in 49BC, " Alea iacta est"("The die is cast"),meaning that a path has been begun from which there is no turning back. While modern-day mathematicians might use dice to simulate random outcomes, many civilisations - paradoxically -understood dice as a medium through which they could divine the will of the god/s of the day. That word divine is pleasingly chosen: it means both 'the quality of being God-like' and 'to predict in a supernatural way', from the Latin divinus, divinare.
'Loaded dice' is a concept you may be familiar with: dice that are fated or expected to give a certain result, related to the latter philosophical idea of predestiny and the more practical mathematical idea of bias. The term comes from the way in which dice can be tampered with to give a particular outcome by 'loading' or 'weighting' the opposite face. One ingenious way of doing this is to drill one of the pips and add a small weight to the hollowed-out inside. Then add a substance like wax, with a melting point close to human body temperature (I imagine chocolate would work just as well, but this feels like something of a waste…). Warming the wax by breathing on the dice or holding it in a closed fist would then be enough to melt the centre and let the weight fall to the bottom face; opening the hand and letting it harden then 'loads' it to fall the same way when rolled. This could be the origin of the idea of blowing on the dice - the superstitious ritual that we see in film and TV (some earlier attempts to load dice may have used sticky substances that unscrupulous gamblers also blew on). You can buy loaded dice (or try heating them a little - but be careful!) and give them to pupils to experiment with: this is the stuff of particularly fascinating, exploratory lessons.
What other types of dice are possible? You may be thinking of the Platonic solids: four-sided (tetrahedral) dice are common (but slightly trickier to label), as are eight-sided (octahedral), twelve-sided (dodecahedral) and twenty-sided (icosahedral). What might they be used for?
Throughout the Middle Ages and all the way into the last century, dice games have been accessible to all strata of society - they are cheap, easily made from common objects, and - unless you cheat - a great leveller for players of different abilities. More recently, gamers have begun to see the random outcomes on different types of dice as ways to model narratives - from determining whether you impress a crowd with your juggling skills to whether you win an epic battle with many thousands of troops. This is the world of Role-Playing Games (RPGs), of which Dungeons and Dragons might be the most well-known. Within a compelling context of storytelling and adventure, you might roll a D20 (20-sided die) to determine the outcome of a conversation with someone, for example. If you are a silver-tongued thief with a high charisma score, you may only need a 7 or above to persuade an unwary child to give up information; but if you're talking to a warrior priest who is known for killing first and asking questions later, the threshold might be 14 or above.
In such games, dice with ten, twelve or twenty possible outcomes give you the flexibility to take into account different starting abilities and their possible increase/decrease throughout the game or campaign. This means you can set thresholds for success or failure that are very flexible and can be easily adapted to the story. Rolling dice also adds a nice element of risk balanced with strategy to critical points in the story, and a good understanding of probability helps but doesn't guarantee success, of course. Neatly, dice can also be used in a completely different way: to track scores (such as health points or resources). I highly recommend trying out a simple role-playing game in the classroom and allowing students to explore the feel of different dice and the limits of their imagination when building a story around them.
Some war games even use fifty- or one-hundred-sided dice, or combine two D10s for the same effect (just use one die for the tens and the other to denote the ones). There are multiple opportunities here for using dice to explore place value and as a way of helping pupils manipulate numbers with concrete representations. For more advanced mathematicians, watching these amazing videos about non-transitive or skewed dice will help them to understand more recent developments in the history of the humble die. Have these simple objects had their day in the classroom? I say 'no dice'.