In my third year of teaching, one of my students said to me, "In Vietnam, our teachers ask us to solve each problem three times; each time, finding a different way to come to the same solution."
This statement, so small at the time, has become a way to define my philosophy of teaching. I hate to show my students algorithms and often try to pretend that their books don't have anything of use for them. Many students learn a way to solve a problem and that is all they have- a method with no understanding of what is happening or why.
This has always been my problem with elementary mathematics education. Instead of having students find their own ways to solve problems and allow them their own methods, students are told that they must add or multiply using one given algorithm. I grew up on this system and was very afraid when I first started teaching because my arithmetic skills were so poor, I knew that the students would have a hard time trusting I knew anything in math.
But as I continued teaching, I watched the different methods that my students used and started to pick up on their tricks to solve problems. I started making my students do their homework problems on the board, not just to get me out of the limelight, but to show methods of solution and ideas that I would have never thought of. My students learn from the varied solutions of others and can look at a problem a peer solves and ask, "Why did you do it like that? Here is what I saw..."
I don't regularly ask my students to solve problems multiple ways but I know that they can. If a student is done early and has a particularly messy solution, I'll ask them to go back and try to solve the problem again. Sometimes, I go around the room and find as many different ways to solve the same problem as possible and have all the students go up and share their different solutions and then we evaluate them as a group.
What my Vietnamese student gave me was not just a teaching strategy but a reminder of the creativity involved in mathematics and beauty of each solution.