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.


  1. I love this post. It is sooooo amazing that you have explored this method of teaching. Kudos x ∞
    (that's kudos times infinity in case it does not show up) :)
    If there were many more educators willing to do that, our current problems with math education would nearly cease to exist. Math is beautiful just the more "traditional" methods of teaching it makes it so scary for so many students.

  2. John, I've never been given "Kudos x ∞ !" :) THANK YOU!!! :)

  3. I have been passionate about math education since 1997 in an algebra class that was being taught by the soccer coach who barely had a teaching degree.
    I was failed in the class for not using borrowing when I showed my work for my final exam.
    That class should have been an easy A for me because I had already taken it 2 years prior in another school.However, this was not the case.
    The teacher proceeded to tell me ,and the rest of the class, that you can not multiply two numbers from left to right and that since I did not use any borrowing I was clearly "cheating". I was even sent to the principals office and given 3 days in school suspension for "arguing" with my teacher and "cheating". I was crushed by this, and it even caused me to not pursue a math degree when I left high school because I was afraid of failure for not conforming to the outdated "machine" approach to math. So to wrap up reading about educators that acknowledge and encourage alternative methods in math, brightens that fire I had so long ago before being crushed by a "terrible" teacher. So this is why the infinite kudos

  4. John, I hope you enter the teaching world and find things have changed. (Or set out to change them yourself!) I have seriously been changed just from teaching Bridges. It's forced me to think about alternate ways to solve problems. I'll have a post up in the next couple days that talks about finding alternative ways to look at math. Hope you visit regularly! I love your enthusiasm!

  5. I am so thankful for your post-it came to me at the perfect time when I've just realized that the way I've been teaching my son math isn't working for him. The Bridges to Mathematics program looks excellent-I checked out the website this morning. One question for you, though-did you have to buy the whole kit in order to get the teacher's guide? Or do you use it without the guide? Thanks for your inspiring blog!

  6. Hi Mindy,
    When I first starting using it, I only had the books. Later I got the whole kit. The site also has a lot of free supplemental materials.
    Thanks for such a nice note!

  7. I admit that I was taught standard way but I was in school for mathematically gifted where everyone was allowed to share their own ways for solving problems. I am curious now if I "ruined" my 5 year old by showing her standard carryover and borrowing. She understood it within 10 minutes, but this is how she normally is with algorythms. Perhaps I should have let her develop her own techniques first.


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