Tens and ones, left to right

One of the kids I work with doesn’t like to carry digits the old-fashioned way when he’s adding multi-digit numbers.  He has his own method.  It frustrates and confuses his parents and invites correction by his teachers, but he still does it, and he doesn’t usually make mistakes.  The first time I watched him do it I was very impressed.

He was adding 74 and 18.  He started by adding the 7 and the 1 (hence the adult dismay).  He wrote an 8 in the tens column of the total (more adult dismay; how were we to know that he knew it might be only a temporary 8?) Then he moved over to the ones, realized that 4 and 8 are 12, and thus couldn’t be recorded in a single column.  So he returned to the tens column, changed the 8 to a 9, and recorded the remaining 2 in the ones column where they belonged.

If those of us who learned to carry tried to do it this way, we’d likely make mistakes, get confused, call it difficult, and beg to go back to the old way.  But what he’s doing makes sense to him, and it demonstrates an understanding of what’s happening: you can’t record anything more than 9 in a single column, so as soon as you hit ten, you have to add to the next column over.  He completely understands what he’s doing, which is more than you can say for many a child (many an adult for that matter) who dutifully memorizes the system of carrying but hasn’t much of a clue why it works.

It’s a good reminder that a child’s apparent inability to perform a task as directed is not necessarily the whole story.  This child understands perfectly well about tens and ones – there’s no way he’d be able to pull off his method if he didn’t.  He’d end up with missing tens or extra digits all over the place.

And who can blame him for wanting to devise a system consistent with the left-to-right progression he’s been used to since he began to learn to read?  To suddenly have to start moving from right to left probably felt awkward. The carrying system is undeniably clean, but if it confuses you because it feels backwards, why not find another way that works but doesn’t feel backwards? This child came up with a way to keep track of things that not only works but demonstrates that he actually understands what he’s doing.  Pretty great.

Addendum: One reader points out that if this child sticks with his homegrown approach when he’s adding bigger and bigger numbers requiring hundreds, thousands, etc., it’ll be an awful lot of work to keep erasing and revising.  To be sure, it will.  I didn’t mean to suggest that his method would be efficient long term; only that it’s providing a helpful entry into an otherwise (for him, at least at this point) confounding system.  If he has a lot of pencil and paper arithmetic ahead of him, which he may or may not; I don’t see anyone doing much by hand anymore, he’ll likely want to switch over at some point to the old-fashioned way or something similarly efficient. In my experience, having the grounding in any method that is based in understanding – however inefficient it may be – can make other methods more accessible.

I’ll also add that I don’t recommend withholding traditional methods from kids just because they might find their own methods early on.  Whether or not a child decides to use a traditional or otherwise well-known approach to something, it’s helpful to know that it’s there, to have knowledge of and access to its potential advantages, and to realize that others may expect it to be done that way.  It’s possible to allow for constructed methods that increase early understanding and to let kids know what else is possible.

Math Practice for the Younger…

I hung on to a daily math practice book from my last classroom teaching job, and it’s proven a good keep.  It’s published by Great Source, and is set up as a quick review for a range of math concepts (place value, fractions and decimals, etc.).  It’s called Practice Counts, and comes in several different grade level versions.  (It can be tricky to find – it’s available on amazon but you have to check with the vendor to be sure you’re getting the level you want; you can also order it through a retail store.  The publisher only sells them several at a time.) Continue reading