Twelve and the real life problems problem

The number 12, not the most esoteric secret of mathematics. Yet through the under appreciated power of the equals sign it can become 6+6, 3*4, 15-3, 36/3 aall before we add in unusual versions of the number 1. 7*12/7, 10+12-10… None of this even starts to help use the number in the world. It can be an office number, the month December, the number of sheep in a field.

Focus on that final one. A farmer with five sheep in one field and seven in another can combine those two numbers to know he has 12 sheep without getting them all in one field to count. This is the heart of mathematics. We can use the equals sign and other tools to generate new facts from old without having to go back to the evidence. We know that putting sheep together in a field and then counting will give the same answer as counting and then adding.


This is actually a very profound idea and a formal version of it is at the heart of category theory1, even considered by many mathematicians, to be abstract nonsense. Yet it is something we do every day. When we use numbers we intuitively take into account the properties we can use. Go back to that sum:


When both fields counted sheep we could combine them. If one counts sheep and one cows then we have a units problem, though we could combine to get 12 animals. Even with the same units we can not always use all the properties. Offices 5 and 7 do not combine to give office 12, May and July do not combine to give December. It is in a real sense five months added to July. Though if we add five months to December we can back to May, which opens up several questions!

I could go into the formal details of these different systems, that is not the point. Instead we should realise that even dealing with something as mathematically nonthreatening as the number 12 there is a huge amount of complexity in how it is used.

This issue is at the real world problem problem. To show how a particular technique is used we must cut away all the translation issues, we must even cut away all the other mathematics until, however important and sincere to begin with the problem feels trivial and fake.

At this point you might hold up your hands in despair. To me this gets to the heart of the teaching of mathematics. There are (at least) two distinct skills:

  • The mechanical system of converting one thing to another using an array of symbols
  • The translation system mapping those moves onto the world.

We can emphasise the mechanical (the approach of most school systems) or we can emphasise the translation (the approach of Conrad Wolfram and others who argue that computers should do the calculation). Personally I sit on the balance. Without the translation your models become just intellectual games. Yet without the models you have nothing to translate to2. The challenge of teaching mathematics is to balance between the two. It is hard especially under all pressures that “things would be simple if…”. Its hard but on occasion mixing the two right can set off a virtuous cycle where advancing in one motivates effort in the other.

1 BACK TO POST My article Category Theory for Designers has a little bit more on this topic.
2 BACK TO POST Just translating to a computer might have potential, especially Keith Devlin’s approach of using computer games to develop intuition. Yet if not careful it can miss the understanding of what the model actually does so becoming a magical incantation.