## a bit fishy...

Around 50 years ago, the biologist Robert May was modelling the population of fish using the following equation called the

**logistic difference equation**:

**Population next year = R x Population this year x (1 – Population this year)**where the population is a percentage and R is a growth factor.

For different values of R, he found that the population values behaved in a strange way, forming the foundations of chaos theory. This article explains his findings in more depth.

This spreadsheet allows you to investigate the behaviour of the logistic difference equation for different initial populations and growth factors.

## chaos in the classroom

Robert May advocated the use of an investigative approach to solving mathematical and scientific problems:

Inspired by this, I wanted my students to investigate chaotic sequences themselves. I gave them the choice of investigating the logistic difference equation or another sequence that appears to behave in a chaotic way, called the Collatz conjecture.

It is generated by the following rules:

This video

Both the students and I found the lesson interesting (if slightly chaotic) with students experimenting and discovering maths for themselves. They were practising important skills; using calculators, trying things out and explaining what they had found – just as May had advocated.

*"The world would be a better place, if every young student were given a pocket calculator and encouraged to play with the logisitic differential equation. Chaos should be taught. It is time to recognize that the standard education of a scientist gives the wrong impression. No matter how elaborate linear mathematics can get, it inevitably misleads scientists about the non-linear world. The mathematical intuition so developed ill equips the student to confront the bizarre behaviour exhibited by the simplest of nonlinear systems. Not only in research, but also in the everyday worlds of politics and economics, we would all be better off if more people realized that simple nonlinear systems do not necessarily possess simple dynamical properties."*Inspired by this, I wanted my students to investigate chaotic sequences themselves. I gave them the choice of investigating the logistic difference equation or another sequence that appears to behave in a chaotic way, called the Collatz conjecture.

It is generated by the following rules:

**Starting with any positive integer**

- If

- If

Now take*n*, form a sequence in the following way:- If

*n*is even, divide by 2 to get a new number*n'*=*n*/ 2- If

*n*is odd, multiply by 3 and add 1 to get a new number*n'*= 3*n*+ 1Now take

*n'*as your new starting number and repeat the process.**For example, if you start with n = 10, you get the sequence**

*10,*5, 16, 8, 4, 2, 1, ...This video

**shows what happened in my year 9 class when we tested it out the two chaotic sequences.**Both the students and I found the lesson interesting (if slightly chaotic) with students experimenting and discovering maths for themselves. They were practising important skills; using calculators, trying things out and explaining what they had found – just as May had advocated.

ps there is currently no proof that the Collatz conjecture always ends at 1.

ps there is currently no proof that the Collatz conjecture always ends at 1.

**John Conway**once stated that ‘Mathematics is not ready for such problems.’