Problem-solving. Solve the problem. How often do you hear a math teacher say that? Whoever decided that the exercises we do in a Math class should be called "problems" has seriously injured math instruction. Are they problems? Is my life any different as a result of a x^2 - 3x + 2 = 0 not being solved for x? Problem solving is not, "Here's a math problem, solve it!" Problem solving involves real and relevant scenarios, not just x's and y's (the m and n when the teacher wants to change it up).
So, give me a real "problem" in a mathematics class. Give me something that's relevant to me. Answer the question of "When am I going to use this?" For may instructors, the only answers to that question are: "In College", "When you go into engineering", "In Business". Relevance is the key to getting kids to "learn" math instead of "doing" math. This is the reason that I enjoy Statistics, as I can get my students to represent relevant situations with my course content. I hear from many of my colleagues that, "Statistics is an easy course to do fun labs and activities in". I don't disagree with this assertion, but there is a whole universe of evidence that exists to show how easy it is to "do fun stuff".
So how do we make it relevant? The best examples I've used with my Calculus class involve gravity and motion. Drop a ball. No wait, have your kids drop/throw a ball. Have a student walk/run/stand still. Roll a Matchbox car across the floor. Drop a ball into a bucket of water. Capture all of these on video and have your students take measurements and data. You'll have instant "fun stuff" for an entire unit on quadratics. Our Algebra II textbook has 1 section of 6 devoted to "Applications of Quadratic Functions" and not one of them mentions motion. My thoughts are that motion should be your entire unit.
Is it really relevant? If students are interested in it, I think it is. If students are invested in their own data collection, it's a step towards relevance. They are solving their own problems, not the ones that come out of the can and onto the worksheet to be "done".
When I look at a standardized test, I see the relevance completely removed from mathematics. Most of these test questions are "Solve/Simplify". They try to crowbar relevance by making a student "Explain your work", but it only perpetuates the "Do the problem, get an answer" that is not relevant at all. "Then I subtracted x from both sides" Why? Why would you do such a thing? Are you illustrating an understanding of Algebra, or simply following procedures?
If you want kids to do a procedure, have them go on YouTube, watch a bunch of how-to videos, and learn the procedure on their own. An entire class period devoted to learning a procedure is a waste of time. If you want them to learn mathematics, they need to interact with it in a relevant context.
By the way, I have been blogging lately by writing my posts in Google Docs and then copying and pasting at home, where my blog is not websensed.