“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.” G.H. Hardy
Technology allows us to do wonderful things with iterated procedures. I tend to use calculators and spreadsheets to start with, as they give a very visual response and a “hands on the wheel” feeling about what is actually happening. Obviously you need to get to something more sophisticated to get a lot of results or generate pretty visual or aural images.
A Note on Software
As I say, spreadsheets and GDCs are great. I am of an age where I still think Logo has a tremendous amount to offer here. All the functions, procedures, variables, recursion, graphics and sound are here. It is not too syntactically complex to turn kids off and simple stuff can be very simple. The Cantor Dust used as the header in this site is just a few lines in Logo. I use mswlogo from Softronic. It is free. The help is great and the musical functions are easy too. I think acslogo works just as well for a mac, but I do not know about its musical functions.
If you want to go further and get them into Java, I really like Processing. It is the easiest way to start Java that I have come across. There are obviously things like Jurtle and Python as wel, but you need kids so confident that they can get an outcome that they can surmount the challenges of the syntax. So, start with Logo (or Scratch then Logo)
A starting activity. As with nearly all activities here, it is written so students can just get on with it. However a more open ended investigative approach might lead to greater rewards in terms of the skills of research, organisation and opportunities fro creativity. You can just say “what happens if you repeat something over and over again?”. Then just keep prodding them to try different functions and starting points, using the sheet as your guide.
Of course this is more fun in two dimensions, but you do need something to draw things for you. The sheet linked to in the title starts you off in Logo, and links to some work in Processing. I am not a teacher of programming, so I cannot share any great expertise. However you can still have fun with low level programming skills and mathematical curiosity.