Addition is really just a short-cut for counting. When adding smaller numbers, like two plus three, it is almost as easy to count to get the answer, as it is to add. However, the larger the numbers get the more difficult it is to use the counting method. This is where the "short-cut" of the addition algorithm comes in handy. Our math curriculum suggests setting up a scenario to help the student "appreciate the method for adding large numbers".
I gave Michael a series of addition problems to solve. Each problem asked him to add together larger and larger numbers. I instructed him that he must represent each problem using blocks and then count the blocks to arrive at the answer.
Mike found the exercise amusing. He enjoyed putting together all the necessary blocks and counting them up to show the answer.
In this set-up scenario, the student is supposed to see that eventually counting becomes an inefficient way to arrive at the total. And when given the final problem, asking them to gather and count two numbers in the thousands, the student is supposed to give you an incredulous look and protest at the enormity of the task.
Not Mike, though.
The prospect of adding 5,342 and 3,254 excited him and he cheerfully got to work counting out blocks. While it is true that given enough time and math manipulatives he could represent the problem and count the answer, he wasn't quite getting the point of the scenario I had set up. Either that, or he had decided that counting blocks all day would be more enjoyable than the other things I had planned.
Exactly, how long was I supposed to let him count out blocks? I decided to push the issue by asking a few pointed questions about our relatively limited supply of blocks and the time it might take to show and count such large numbers.
King jumped in to help by commenting,
Dude, you're still only up to five hundred!
Finally, Mike reluctantly admitted that, even though it was possible to arrive at the correct answer by way of counting blocks, there might be a better way to use our time.
So much for the appreciation of addition.
The next day I put this problem on the board:
I led him through adding each column, starting with the ones. When we were done I asked him to read the problem again, including the number we had just created below it.
Suddenly, something struck him,
Wait! Does that number equal those two numbers added together?! Wow! That's so cool!
Without any elaborately set-up scenarios, Mike had arrived at a true appreciation for addition.
I had found a new appreciation for what a privilege it is to be my children's teacher. It is moments like these, where I have the opportunity to witness the excitement and joy of discovery, that make all the hard work of teaching my children worth while.