Frost started his schooling at an alternative school which emphasized math creativity and understanding. Kids sorted marbles into bottles representing 10s and 100s and counted macaroni threaded on yarn. Frost was encouraged to form his own methods for adding and solving problems and generally thinking about math. He did. He figured out ways to move numbers around long before he was taught the established ways. Even my Dad was impressed by his "mental math" with large numbers.
Fast forward to 5th Grade and these techniques are getting in the way.
He is doing story problems with decimal long division and multiplication. The typical questions are about people buying items with sales tax and discounts. They involve finding the new price by calculating 9.25% of $19.99 and stuff like that. Josh and I both recognized that these problems are easily solved by doing the grunt-work of math - the layout, the algorithm, the carrying and solving.
Frost resents this. He hates doing the straight long multiplication. He breaks things up into funny simpler functions and adds them up - typically making mistakes along the way. About half the time he solves the whole problem in his head correctly. The other half of the time they are incorrect and either show no working method (ie, are just plain wrong) or use a contorted sequence of logical steps that he has devised.
NOT the algorithm.
Today, I dropped into the 'advanced learning' school to pick up some books. I met a 4th Grade girl working in the corridor. She came up to me in some excitement and said "look how much work this problem is taking me!" Indeed, her lined page was covered on both sides with detailed, neatly laid out sums. They all seemed to be a huge number subtracting 20.
"What on earth are you doing?" I asked.
She was a bit confused by my lack of enthusiasm.
"I am solving a problem! This is how much MATH IT TAKES!!!" she asserted, waving the page at me.
"What's the problem?" I wondered, secretly thinking that we never did such long sums in 4th grade and WTF was she doing?
"Oh, these people have $10,000 and we need to know how many weeks it will take them to spend it all if they spend $20 a week. SOOO I am subtracting $20 each time. Look!"
I look, and indeed all the multitude of sums are subtractions, $10,000 - $20 = $99 980 $99 980-$20= for two whole pages!!!!
"But why do it that way?" I asked. "Why not divide? Just divide it by 20!"
"No, I am using SUBTRACTION!" She affirms with mistaken confidence.
"What about dividing by 2?" I ask, freaking out politely. This kid is in advanced learning 4th Grade, she should rebel if being asked to break down $10, 000 by $20 doing the dum sum 500 times. Even if she can't divide 10,000 by 20, intelligence demands that she rebel!
But she doesn't. She prances off waving her pages of sums, committed to solving the problem using the alogrithm du jour, subtraction.
So, I don't want Frost to be like that. I want him to say "this is dumb, there must be an easier way" but I also want him to listen to me when I tell him that he is in the hard way, that sometimes you have to exercise the brain to show it how to make something easy (like the classification of mushrooms or the mechanics of algebra).
Often we have to do a bit of hard work to get to the easy path.
Frost does not believe me, yet.