I love reading about the ability to go deep and wide in math, instead of just plowing through the material faster. We have been staring at the Calculus Trap (without knowing that it had a formal name!) for some time now. But I wanted to make a few comments.
In one of the gifted books that I have read in the last several months, (I can't keep them all straight now, but I think it may have been Karen Rogers?), the author describes the effect of math drudgery, or 'drill and kill", on the gifted student. She (if I am correct on the reference?) stated that mathematically-inclined gifted kids learn the concepts in math exceptionally fast, which is no surprise. But she then stated that with repeated drilling of the same topic, gifted kids actually unlearn the material. It was hypothesized that after the first or second time running through the same material, gifted kids stop paying attention and their scores for assessing accuracy of math facts go down rapidly. So drilling a gifted kid in math is in fact detrimental to their mastery of the subject.
That said, I would also argue that many of the parents on this board, myself included, have posted that their child would rather learn the big, abstract concepts in math rather than do the more mundane calculations that allow you to connect point A to point B. This is worrisome, at least to me with a fairly young kid (okay, at age 8 he is not as young as I think he is!). He needs to be able or willing to focus on the drudgery in order to do more advanced calculations later. Not all of math is built on these big, exciting leaps that seem to fuel the gifted brain.
The problem that I see is that you have to keep them interested enough in the topic by dangling the next big leap under their noses so that they are willing to do the drudgery that fills in the details. If they race from big idea to big idea, then they are missing the structure that holds the whole house of cards together. And if they get so stuck on the minute structure that they miss seeing the grand picture of the whole house, then they dig in their heels and lose that spark of learning.
I think with our kids the pitfall of missing the structure is much, much less than losing the spark. We know our kids and we know what they are capable of and how fast they can go. The schools are focused solely on the drudgery that fills in the gaps in the structure. Their achievement tests depend on the kids mastering the drudgery, their school report card (a la NCLB) depends on it, and frankly they can't imagine any instance where going over and over the drudgery one more time is not a good thing.
I guess the only way around this is to get the school to acknowledge that once the child has mastered the structure or details of a particular lesson, then it is in the best interest of the gifted child to be allowed to move forward. They need to do a certain amount of drudgery in order to make sure that they have mastered the material, as well as to build character and learn perseverance. But for goodness sakes, then they need to move forward!!
And if anyone knows of a way to get the schools to understand this rather simple concept, then please bottle it and sell it. I would be willing to purchase great, big jugs of it!
We are currently stuck with lots of instances of repeating the previous year's drudgery (as if 4th grade drill and kill wasn't bad enough... the teacher is repeating most of last year's drudgery, just to make sure that all of the kids have it.... yawn!), and balancing it with any afterschooling in math that I can get my hands on. (hence the post on Mandelbrots sets). This is surely a kluge job, and may not last for long.
When (not if) DS reaches the end of his patience for the slow pace of 4th grade, we will have to start fighting for another solution.
I hear you, 'Neato, and will probably be in your shoes shortly. Any suggestions for the deep and wide aspect of math would be greatly appreciated, as it may stave off inevitable battle of subject acceleration.
