The more I learn about the maths in different countries the more amazed I am that we have assumed for so long that math is the same the world over. But that's another story.
The question I have been wrestling with for quite some time now is why we still teach the four basic algorithms as illustrated above. The procedures involved in teaching all four algorithms to the level expected by sixth grade still consumes a large part of the elementary school curriculum even though almost everyone these days has a personal electronic device that will do the calculation in a nanosecond. The saddest part of all is that so many students who are able to complete the procedures have little idea about when to use them and even worse, many forget how to do it because, in real life, they never have to do it. When was the last time you worked out with a pencil 398 divided by 72?
Above is a traditional addition algorithm although for some inexplicable reason the smaller number is on top. (remember how you were told to put the largest one on top as you would need to do this in subtraction?) The example above also has the wonderfully anachronistic "carry" as a command of what to do with the ten ones which magically become just 1 (interestingly ten 1s become 1 ten which changes the word 'ten from a noun to an adjective - the linguistic essence of regrouping).
So think about those two numbers 15 and 29. In what relationship could they be?
How about two groups of people, one of 15 and one of 29 being joined together into one group. Maybe combining two college classes because a professor is unwell.
How about a large gathering of people, such as a school staff, in which you know there are 29 women and 15 men and you want to know the total number of people. There are no separate groups; this is what we call part-part-whole.
How about two bags of candies? One bag contains 29 candies and the other bag contains 15 more than the first bag? This time we start with two groups and one of the numbers is not a group at all but the difference. Or one bag contains 15 candies and the other contains 29 more?
How about wanting to know how many free tickets you started with when you have given 15 away and now only have 29. You can even use addition to find the answers to problems that solve separation.
We could so easily set up the relationship like this; 15 + 29 = ? This would help children develop the real sense of a numerical equation along with the most important thing in algebra; the idea that the = sign means "is the same as".
If only we could spend our time developing children's number sense and their ability to recognize mathematical relationships then use a calculator for the drudgery part instead of the endless hours we waste with mindless, boring and tedious algorithms.