If you have a friend with you please ask them to place a small object on the floor, somewhere in the room that you’re in. Now go and pick it up. Easy? I should hope so!
Next close your eyes and, keeping your eyes shut, get your friend to place the same object at a different location within the room. Ask your friend to direct you to where the object is located (please be very careful and make sure there is nothing dangerous in the room such as a fireplace or sharp objects before you do this). Your friend must give you precise instructions – you mustn’t peek or cheat in any way!
Once you’ve located the object repeat this by placing the object on the floor yourself and giving your friend instructions as to how to reach it. I hope you’ll agree that this is not so easy! You have to give (or receive) very precise instructions and this can be very challenging.
Now let’s try something even trickier.
Decision Maths: Shoelace Task
If you have shoelaces on your shoes please undo them, pause for a few seconds and then retie them. Again I hope you find this pretty easy! Please get a friend to join you at the computer (or email a friend instead – preferably one with a sense of humour as otherwise they may find the email a bit odd!) and ask them to also untie their laces and then retie them but, and this is where things get harder, instead of just tying up their laces as they would do normally, you have to write out a set of instructions telling them how to do up their shoelaces. They must only do what you tell them to do and your instructions must be very clear, simple and specific (for instance you can’t just say ‘form a bow with the laces’ – you have to tell them what a bow is and how to form one!).
The idea behind these two examples is to illustrate how difficult it can be to write down instructions explaining how to perform even very simple tasks. For example, telling someone how to tie shoelaces is very difficult – as I hope you will agree. Yet, if that is true, how come pretty much everyone knows how to do it?
The key here is that when you were taught how to tie shoelaces you weren’t told how to do it, you were shown how to do it, and this is very different. Showing someone how to do something is much easier than telling them how to do it, hence a lot of what we have all learnt has come from seeing how things are done first – just think of what happens in school; isn’t it always easier to learn something once your teacher has shown you how to do it?
But what happens if I want to teach a computer something; how can I possibly ‘show’ a computer what I need it to do? I can’t. I have to tell the computer what to do and this means that I need to be able to write down very precise instructions telling the computer how to solve whatever task I want it to do. This leads us to the branch of mathematics known as DECISION MATHS, which you may learn about if you study A Level Maths.
Decision Maths involves looking at problems (many of which do not look like traditional maths problems) and trying to find out whether or not these problems can be solved by just following a set of instructions (and finding an appropriate set of instructions to use). Once a suitable set of instructions has been found – these instructions are known as an ALGORITHM – a computer can be programmed to follow this algorithm and hence we can get the computer to solve this type of problem for us. Algorithms are used to solve a wide range of problems, from the very complicated (such as finding the cheapest way to link lots of cities by a high-speed railway system) to the more mundane (such as rewriting a set of numbers into ascending or descending order). Now, writing a set of numbers in ascending order is an incredibly simple thing for you and I to do but for even this basic task it is much harder to write down a set of instructions (an algorithm) so that a computer is capable of doing it.
To illustrate this, I would like you and your friend to both write down your age, the year, the number of brothers and sisters you have, the number of countries you’ve visited and your favourite number. Then I would like you to rewrite your list in ascending order (ascending order means that you start with the lowest number, then the second lowest number, etc). For me this gives the numbers 30, 2009, 3, 34 and 1; and I can easily rewrite these numbers in ascending order as 1, 3, 30, 34 and 2009.
Now see if you can give your friend clear instructions explaining how to rewrite their list of numbers into ascending order (again, your instructions must be very clear and simple – you cannot just say ‘write down the smallest number first then the second smallest number, etc’ as you need to tell them how to find the smallest number, the second smallest number, etc!). Try to see if you and your friend can come up with an algorithm that will always correctly order a set of numbers (you can follow the link – http://en.wikipedia.org/wiki/Bubble_sort – to see some examples of sorting algorithms) and in a future blog I’ll thoroughly describe one common example – the BUBBLE SORT ALGORITHM (this particular algorithm is covered at A Level in the module D1).
Decision Maths helps us to solve a wide variety of problems such as working out the best way to design a rail or road network, telling a robot how to move, calculating how much oil can be piped from an oil-producer (such as Russia) to an oil-importer (such as the UK), determining the quickest way to construct a new building, and many other important tasks.
The key to the success of Decision Maths is that by designing algorithms to solve such problems for us, we can get computers to do all the hard work and find the answers more efficiently and more accurately than we could do ourselves!
By Bellerbys Maths Tutor, Dr John McDarby
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