Subtractions and Decomposing Numbers

I am a high school math / science teacher and sometimes I have a hard time understanding how to teach the younger students to understand / learn math concepts.

For instance, I was practicing subtraction with a young student (eight or nine years old). I realize that I could give him many subtraction problems and see how he does with that. The more he does, the better he gets, right? Isn't that the theory - practice makes perfect?

But when I was doing this with him, he would always get stumped with subtracting the larger numbers. Since he took his time, I started to analyze what I did to accomplish the subtraction. And then I saw what he did.

For instance, there would be a subtraction of 11 - 8. He would start counting backwards 11, 10, 9, 8,.... This would take forever (in my mind) when this simple subtraction should be an easy task for any grade 4 kid. Counting backwards is the really long way to go. It will get the right answer, but there are so many better ways of subtracting than counting backwards.

I realized that this kid won't ever think of a different way to do these types of problems, even if he gets 100 subtraction problems every day. If he did get 100 problems every day, he might get really good at counting backwards, and this might lead him to get faster at subtraction. But this kid is not one to do 100 subtraction problems every day. And why should he anyway? He just needs to learn a better way of subtracting...

A kid that enjoys math, enjoys finding patterns, has a knack for these kinds of things would find a better way to do the 11-8 subtraction (even counting forwards from 8 to 11 is faster than counting backwards).

For instance, when I was a kid, I don't think anyone showed me any real methods on how to subtract fast. Even though I remember doing a million subtraction problems, I never remember having a lesson about it. Everything I do in my head (with basic math) I figured out on my own.

I cut the number into pieces: 11 is 10 and 1. Then I subtract 8 off of the 10... which is done instantly, because I know automatically that 8 + 2 is 10. So 10-8=2 is a given.... then there is that 1 left behind... leading to the answer of 3.

And my whole thought process takes very very very little time, compared to counting backwards. In fact, I would call my method instantaneous. However, I don't know if I used this process when I was a kid myself, or did I develope it in my later years of school, in order to make my life easier in math class.

Then I decided to stop the insanity of giving my student 100 subtraction problems, and instead, I decided to teach my cool "decomposing numbers" method. I quickly developed a strategy how to teach him the method (I'm sure it's not the best method, as I only had a few seconds to think about it). I figured the most important thing to use my method is to know the pairs of numbers that add up to 10 (like 3 and 7, and 2 and 8 etc.). I couldn't believe that he didn't know those by heart. He was actually struggling quite a bit with finding the the two numbers. As I said, I am not good at getting a feel for what children should know at what age, but I thought that by grade 4 children have a feel for adding up to 10.

I started by showing him fingers (for instance 7 fingers) and asked him how many were not showing... trying to make him think visually. Once he got that, we practiced the pairs for a few minutes. He got really good after a short while, but I think that this only got into his short term memory, and the next time I see him, he won't remember.

Next I told him my method of decomposing the number into 10 and something else... for instance 12 would be 10 and 2. (Even 8 can be decomposed into 10 less 2.) Then I told him to use his "pairs of numbers that add up to 10" to do all the subtraction. We practiced a bit, and I think he got it. I don't know if he will use it again. In fact, I would bet that if I ask him next week to do a subtraction problem, he'll go back to his counting backwards method.

I need him to realize that his method is very slow, and that he needs to change it. But initially my method will also be very slow for him, because he needs to think about it: its brand new, and he doesn't even know his pairs of numbers that add up to 10 off by heart. Plus my methodology of teaching him wasn't too good, since I just thought of it right on the spot.

This is a bit of an issue: How do I overcome his method, and convince him of using a new one? He's been using "counting backwards" since he learned subtraction, and it's worked for him up to now. Why should he bother with my method?

I'm sure if I made some sort of game out of it, or a cool way of practicing it, this would get him thinking in the right way, and he would change methods even without me telling him to. So either I can find something cool to use to practice this with him, or I can invent a good way of teaching this common method.

I'm sure the resource is out there somewhere, and I'll find it soon enough. But for now the student's mother wants me to work on division with him, and let him be with the subtraction...the life of a tutor.

Update: I found a great site to teach little kids "Magic Numbers" - i.e. the pairs of numbers that add to 10. It starts with a little piggy cartoon, but then has descriptions of many awesome and creative games that can be played with kids and some worksheets that can help practice these important number pairs: Chapter One of Magic Math. Or you can download all the chapters here:Magic Math.

Additional Resources:

• Mr. Base Ten Invents Mathematics: This intriguing story provides mental models that teach mathematical concepts at the foundational level.
• Guided Math: A Framework for Mathematics Instruction: Use a practical approach to teaching mathematics that integrates proven literacy strategies for effective instruction.
• It Makes Sense!: Using Ten-frames to Build Number Sense, Grades K-2