Are percentages that basic?

This blog entry is really just designed to ask the question above and not to answer it. The hierarchy of mathematical sub-topics has always fascinated me and I have never really been clear on what I would view as 'the basics'. Equally I have never really been clear on why fractions and percentages often fall into that category when defined by people in discussion. In fact, students' failings with percentages appear to upset fellow teachers regularly and we as maths teachers are often asked by colleagues 'When do you do percentages?' and 'Why can't your students do them in my lessons?' and questions like these. At the risk of of being irritating I might often respond by asking, 'What do you mean 'do' percentages?'. Almost like probability, percentages are a topic that can escalate from simple to difficult very quickly indeed. How about the following classic problem,

'Start with 2 glasses of wine of equal size, one red, one white. Take a 10% sample from one of the glasses and put it in the other. Now extract the same size sample from the second glass and put it back in the first. What percentage of each glass is white wine?'

That problem can be developed in many different ways very quickly and all of a sudden, many of the people who refer to percentages as basic are struggling. The aim is of course not to make people struggle, but to stop and think about the whole idea of percentages in a bit more depth. Many times I have started lessons on percentages by talking about the 10 % pay rise followed by the 10% pay cut and the associated counter intuitive response. Repeated percentage change, percentage error, reverse percentages and so on are some of the most difficult things that happen in my classroom. I know of students who can apply differential calculus in context, but come unstuck with reverse percentages or in a discussion about why 'Jedi Knight' is the fastest growing religion in the world according percentage growth between surveys. So I am not sure that percentages should be generalised as mathematical basics.

This view was reinforced this week when I read this article,  Poker Machine Maths, about poker machines in Australia. In short, the law states that poker machines should give a return of between 85 and 90%. A lady gambles $300 in a day at the machine and walks away with nothing, but the machine stayed within the law. Follow your instinctive reaction to that statement then read the article. Then, if you really fancy a challenge, imagine how you explain that to a class of your students and the other mathematics that is involved with this problem?