The most creativity in teaching comes not from finding good applications, but in finding good mathematical examples. Without frightening people away from this blog, I can’t give the examples I actually use, but when I walked to work this morning I came up with a similar example for easier math. Suppose I wanted to reinforce the multiplication tables and felt all of my students knew 2 times a single digit, but many did not know other values. If I gave the problems 2 x 7, 3 x 7, and 4 x 7 right after each other, I might get them to understand that all one has to do is add 7. If I then gave 2 x 8, 3 x 8, and 4 x 8 it would not only reinforce this, but some of them might understand that 4 x 8 is 4 more than 4 x 7. Of course I want them to memorize the multiplication tables, but I also want them to understand them.
I grade homework, and although most of the time it is tedious, it helps me learn what kind of mistakes students make. If they confuse two concepts, I make those two concepts consecutive problems. If they can’t do complicated, multistep problems, I break the problems into steps, and then afterwards give them the problems without breaking it down. At the end of the semester I make a worksheet of problems we are currently studying, but throw in some problems from the beginning of the semester.
I emphasize what is difficult but important. Figuring out what that is, requires creativity.