Since I don't have a job yet, I've been doing a lot of extra reading about math pedagogy, wishing I had a class to practice things on. The most exciting I've found so far is Teaching Mathematics through Problem Solving, which unfortunately is out of print (but available used from Amazon). There is a version for PreK-6 and one for 6-12. My first attempt to buy it brought me the PreK-6 book, which I read until the 6-12 arrived today, so I have a head-start on the concepts I'll be reading about. But I thought I'd write a little about my ideas about problematizing math before I get into the book.
My father studied mechanical engineering in the 1930's at Stevens Institute of Technology in Hoboken, NJ, which he always felt was an excellent education. He used to tell me that they had not tried to teach him a lot of facts and formulas (which of course would often be out of date before he had a chance to use them) but how to find the facts and derive the formulas. My college physics professor 30 years later also impressed upon us that we shouldn't learn a bunch of formulas, but instead understand the concepts so we could set up the problems without formulas. With a formula you just have to plug in a few numbers and get answers, but if you don't understand what you're doing you have no way to know if the result it reasonable, or if you, for example, used the wrong units.
From the experience of watching my children grow up, I know that they were more willing to accept facts, rules, whatever, if they had discovered them themselves. In fact, I wrote in my "Mission Statement" for a class this summer:
Through experience with my own children, I know that the younger generation does not want to be fed with my knowledge and experience. Young people learn only what they think they need to learn. Furthermore, they want to experience life themselves, not vicariously through their elders. Any other “learning” remains in short-term memory and can rarely be utilized in their explorations of life. I believe that I can best help young people select what to learn by exposing them to own my passion for learning and exploring; I want to encourage them to maintain their childhood curiosity, rather than to suppress it.I recall that my son usually wanted to do things his own way, even if that way was much more difficult, including climbing up a steep incline instead of taking the stairs. (Of course there were other times when he was feeling lazy that he wanted me to do things for him...)
We get stronger when we do things, we get better at things when we do them often, particularly if we think about how we are doing them. That's how we learn skills. Kids know all about understanding. That's what they spent the first 5 years of their lives doing, mostly without our help, because they had to figure it out on their own. We don't want them to stop trying to understand when they get to school. Skills aren't enough, and can be forgotten. Understanding can be recalled when needed.
Education has had a variety of methods through the years. Socrates had figured out that people needed to figure things out themselves, way back then! But somewhere along the line there's always some know-it-all who figures s/he knows the best way how to do something and wants to save others the difficulty of having to figure it out themselves, or maybe the tragedy of never figuring it out. We all have been know-it-alls at some point or other. (Like when talking with someone who has an opposing view on some topic dear to our heart. Of course they're wrong and we need to make them understand why!) I remember my (then-)husband trying to teach my son how to crawl(!) Why couldn't he figure out how to move one arm forward, not backwards?
I read a really telling example in another book recently about famous environmentalists. One of them as a child had found a couple of caterpillars and followed their life-cycle. After watching the first pupa open to reveal a butterfly struggling to get out, he decided to help the other butterfly, so it didn't have to struggle. But that one never learned how to fly. It was too weak, because it didn't have to struggle.
So by problematizing math, we make students struggle (a little) to figure things out rather than telling them how to do things. We give them a new problem based on knowledge they have already figured out and understood, and let them figure out how to solve it. If different students come up with different methods (resulting in both correct and incorrect answers) we let the students reflect on the methods so that they can decide which ones are most elegant (I love that mathematical term!) are easiest to understand and can be varied to solve other problems, as well as how not to fall into pitfalls that produce the incorrect answer. Then the students can own the methods they understand rather than the steps and skills we present to them. They learn how to understand, rather than formulas to plug numbers into.
Unfortunately there are many who think we should be taking the problem out of math, to make it easier some how. (In my practice teaching this summer we never presented a single word problem, much less have the students figure things out themselves. It ended up being a lot more difficult getting them to remember stuff than if we'd taken the time to help them understand.) There are others who think we should make math "fun." But isn't it more fun to figure things out yourself? Isn't it more inspiring. Don't you remember those things better, and can't you use them in other situations?
I just noticed that one of the editors of this book is an author in a new series Core-Plus Mathematics for which I just bought the (relatively inexpensive) teachers' guide,
My mind is already working out ways to used problematical math in the classroom. I hope I have a classroom where I can use my ideas soon!
- ... a standards-based, four-year integrated series covering the same mathematics concepts students learn in the Algebra 1-Geometry-Algebra 2-Precalculus sequence.
- Concepts from algebra, geometry, probability, and statistics are integrated, and the mathematics is developed using context-centered investigations.
- Developed by the CORE-Plus Math Project at Western Michigan University with funding from the National Science Foundation (NSF), Core-Plus Mathematics is written for all students to be successful in mathematics.