The analysis and simulation of spiral growth in plants integrates algebra and trigonometry in a botanical setting. When the ideas presented here are used in a mathematics classroom/computer lab, students can better understand how basic assumptions about plant growth lead to the golden ratio and how the use of circular functions leads to accurate spiral growth simulations. Using Mathematica, students can experiment, explore and discover how varying a small number of parameters leads to a wide variety of simulated plant forms. The Mathematica code presented here can be easily modified for experimentation and easily rewritten for use with other maths software packages such as Maple and Matlab.