Monday, March 19, 2012
Math Myth: Borrowing is the Only Way
Subtitle: How I "Borrowed" Trouble and Have "Carried" It With Me...
I recently wrote an entry that included a link to a piece of research on "The Harmful Effects of Carrying and Borrowing." This has been on my mind a lot this past year. Here's why...
The math curriculum I use, Bridges in Mathematics, does a tremendous job at having kids explore various ways to solve problems. The way the lessons are set up, kids develop their own strategies, share their thinking with others, and refine their ideas as they become and more and more efficient at math. I was NOT taught this way and it's been amazing to watch my students develop strategies that are far superior (meaningful, more efficient, etc) than the algorithms I was forced to memorize.
This year I've been working with a 9-year-old. He's been able to use a variety of strategies to solve multi-digit addition and subtraction problems. Here's an example of one way that he solved a multi-digit subtraction problem. (Taken from a Bridges Grade 3 assessment.)
1300-600 = 700
700+49 = 749
749-75 = 674
I asked him how he did the last step. He said:
750-75 = 675
675-1 = 674
The strategy shows number sense and comfort with place value. As a child I was told that the correct way to do this problem was by borrowing. It made no sense. No one bothered to familiarize me with place value and number sense to give the problem any context. I had no options except to follow the rules. So why, then, would I mess up this child's system by giving him rules?
I did, you know. Here's what happened...
After being exposed to a wide variety of strategies and showing marvelous strength in his own knowledge, I introduced the standard algorithm. Why? Standards specify that kids must know how to do the standard algorithm along with other strategies. So we did it.
It seriously messed him up. The harder I tried to teach him the standard algorithm, the less he maintained a grasp on the meaning he'd constructed for himself. I taught the standard algorithm using both manipulatives and numbers, so it wasn't like I taught it completely out of context. We looked at how the numbers had to be broken apart (carrying/borrowing) with the manipulatives.
But it made no sense to him.
So I abandoned the standard algorithm. Actually abandoned all multi-digit addition and subtraction for awhile because he seemed so defeated. I seriously thought I'd ruined the kid. When we came back to it, we did a lot of work with manipulatives and I let him do it HIS WAY. Lo and behold, it came back to him. Stronger now than before.
I do still occasionally show him the standard algorithm. But only after extensive time with the exploration of other strategies. And only after he seems very, very comfortable in his own methods. I have abandoned telling him to "do it this way."
Cause "this way" may not be "his way." And "this way" may actually do more harm than good.
I'd love to hear about your experiences.