Math: More Getting What You Need
Most of you will probably agree that government schools--and schools in general--are shortchanging our children, particularly in the areas of math, science, and in world view perspectives. For now, we will concentrate more on math. As a follow-on to my previous post, I wanted to explain that reading wikipedia-math for 15-30 minutes daily is not just for adults who want to brush up and round off their math knowledge. It might be particularly helpful for those young brains that are in the middle of the pruning process. A good parent will lead his children to such resources, and help the child to utilise them, while the children are still young enough to have faith in the parents' judgment.
Here is a good essay on "how to read mathematics." It is meant to aid in reading math journal articles and texts, but it can also be adapted to the wikipedia and mathworld approach.
A reading protocol is a set of strategies that a reader must use in order to benefit fully from reading the text. Poetry calls for a different set of strategies than fiction, and fiction a different set than non-fiction. It would be ridiculous to read fiction and ask oneself what is the author's source for the assertion that the hero is blond and tanned; it would be wrong to read non-fiction and not ask such a question. This reading protocol extends to a viewing or listening protocol in art and music. Indeed, much of the introductory course material in literature, music and art is spent teaching these protocols.
Mathematics has a reading protocol all its own, and just as we teach students to read literature, we should teach them to read mathematics.This article categorizes some of the strategies for a mathematics reading protocol. I am sure my readers will think of many strategies that I missed. The point is that there *is* such a protocol, that we all know and use it, and that we should make an attempt to share the secret with our students.
....."Reading Mathematics is not at all a linear experience ...Understanding the text requires cross references, scanning, pausing and revisiting" (ibid page 16).
Don't assume that understanding each phrase, will enable you to understand the whole idea. This is like trying to see a painting by staring at each square inch of it from the distance of your nose. You will get the detail, texture and style but miss the picture completely. A math article has a story! Try to see what the story is before you delve into the details. You can go in for a closer look once you have a framework to fill with details, just as you might reread a novel.
....Mathematics says a lot with a little. The reader must participate! At every stage, he must decide whether or not the idea presented was clear. Why is it true? Do I really believe it? Could I convince someone else that it is true? Why didn't the author use a different argument? Do I have a better argument or method of explaining the idea? Why didn't the author explain it the way that I understand it? Is my way wrong? Do I really get the idea? Am I missing some subtlety? Did this author miss a subtlety? If I still can't understand the point, perhaps I can understand a similar but simpler idea? Which simpler idea? Is it really necessary to understand the idea? Perhaps I will just accept this point without understanding the details? Perhaps, my understanding of the whole story will not suffer from this?
Putting too little effort into this participation, is like reading a novel without concentrating. After half an hour, you wake up to realize the pages have turned, but you have been day dreaming and don't remember a thing you read.
....Reading mathematics too quickly, results in frustration. A half hour of concentration in a novel buys you 20-60 pages with full comprehension (depending on how experienced you are at reading novels). The same half hour in a math article buys you 0-3 lines (depending on how experienced you are at reading mathematics). There is no substitute for work and time. You can speed up your math reading skill by practicing, but be careful. Like any skill, trying too much too fast can set you back and kill your motivation. Imagine trying to do an hour of high energy aerobics if you have not worked out in two years. You may make it through the first class, but you are not likely to come back. The frustration from seeing the experienced class members effortlessly do twice as much as you, while you moan the whole next day from soreness, is too much to take.
And so on. Scan through the essay as you can, then apply the approach to your next 15 minute math session. Persistence plus good technique. It takes time to learn the right technique for you--the best approach--but persistence is there for everyone, for just a little willpower.
You might also want to check out something called Visual Math. Math can be very beautiful, visually and esthetically. Here are some links to visual geometry.
Finally, here is a math reference website, to add to all the others in my previous posting, and the ones on the sidebar.
Combining the information in this posting and the previous one, you may be in a better postion to compensate for any deficits in your own math education, and to prevent large gaps from forming in your children's math educations. Math is not all they will need to know, but it will be a key part.
Labels: science teaching