When we were young, we learned and studied patterns. When we start learning arithmetic and higher math, we (tradtionally) learn sets of instructions. I find this to be 99% wrong. There is a time and place for learning a set of instructions, but not that often.
Math (and/or nature?) is nothing but patterns. We, as humans, learn from patterns. Why not present all math as patterns and let the students discover the "instructions?"
Many of my lessons are progressed by asking, "What is the pattern?" I set my students up with the pattern. They do the work, they look back at the work, and should come to a conclusion about at least one pattern from their work. For example:
a. (x + 3)(x - 4) b. (y -5) (y + 7) etc...
What patterns do you notice? How can we use this info to work backwards, or to factor?
The students come up with the rules/patterns/steps/etc. for factoring.
I have found my students are more engaged and excited about math since I have been teaching this way. Let me know what you think or find out.
And I cannot leave this post without sharing a short clip from one of my favorite movies and directors. Daren Aronofsky's "Pi" is an outstanding movie, especially for a first full length film. Enjoy!
Watch a short clip here