There are several different definitions of "pattern"; pattern as a design, pattern as a model or something to be copied, and pattern as a regular sequence or set of events. It is this last definition that is most relevant here. If we can help children see patterns between different mathematical entities it will help them remember, recall and make sense of what they are learning. We can do this from the earliest stages when children learn to count by 2s, 5s and 10s. What makes this type of activity even more worth while is to count by 5s starting at 3 or count by 10s starting at 7. Whenever i do this with my students I can see them initially stumbling and going slowly. Then, as they see the pattern emerge they speed up and end up rattling the number sequence off.
Sierpinski triangle, a classic fractal, as well as in Pascall's Triangle.
Here's another amazing number pattern. Add 1+2 (=3)+3 (=6)+4 (=10) +5 (=15) +6 (=21) +7 (=28) +8(=36). These are called triangular number because they make triangles (imagine 1 + 2 next to each other like steps etc). Now add consecutive triangular numbers together and what do you get? Yes, a square number. Imagine turning 3 upside down and fitting it together with 6 to make 9.
Math is, indeed, the science of pattern.