- procedure without connections
- procedure with connections
By "math" the author means solving problems for which there is no given procedure and can be approached in multiple ways.
My question is whether there is a strong correlation between the level of learning and the impact on our long term goals for our students. I am also curious about the long term goals of the students themselves. So today I wrote down three items on the board and asked my students which was most important to them:
- Preparing for future math and science classes.
- Developing problem solving skills that will be useful in other areas of life.
- Gaining insight that will help them understand deep and fascinating concepts of math and science.
In my Geometry class, the overwhelming favorite was the second option. In my Algebra II class, the first one was the clear winner. I chose these three goals to represent three different conceptions of the usefulness of a math education. The first represents the idea that math is primarily a way to prove to society that you are intelligent. If you learn math, you can get good grades now and in future math classes, and the higher classes are important because if you get good grades you will have a good chance of getting into the college of your choice.
The second option represents the idea that mathematical knowledge can transfer to other disciplines. Most people don't become mathematicians (or scientists or engineers), but perhaps knowing some math may help them in some other endeavor.
The third option represents knowing math for its own sake, because it is interesting and beautiful and sheds light on the nature of the world we live in. Not many students chose this option in either class.
My guess is that most math teachers would select the second option as the most important for a students in a high school math class. However, many of us probably teach as if the first objective is primary. This is because the curriculum is designed that way. We (as in, the state) have selected a bunch of objectives that we think students need to master to be "college ready", that is, to prepare for the next level of math and science classes. In order to make sure our students are prepared for these classes we rush through a textbook that organizes and isolates each concept in a very clean and logical manner.
I suppose one way of putting this is that our curriculum seems to push in the direction of presenting a lot of "memorization" and "procedure without connection" tasks in order to cover everything we need to cover to be prepared for college. Of course, the idea that we need to have a whole catalogue of neatly separated mathematical procedures at hand in order to succeed in college may be erroneous. It might even be better if students forgot everything they learn in math except for some broad problem solving strategies.
In any case, the aesthetic value of pure math seems to be pretty far down the list for both students and curriculum makers.