James Tanton is one of the most insightful teachers of teachers of math I’ve come across in a long time.
In this video and this essay, he explains how some parents feel mystified by some of the newer style of math problems that children are bringing to their parents for help. Clearly, some of the problems (like the 32-12 one which fills up an entire page in a child’s workbook) are being done in a tortured, time-wasting method. However, if you take a more difficult problem, such as 103 – 87, you can do this completely in your head by adding small increments (MENTALLY) to the 87 until you get to 103.
This is a reasonable thing to do, much like people count out (or used to count out) change into a customer’s hand.
In this case, from 87 to 90 (which is a much ‘nicer’ number than 87) we need to go up by 3.
Then from 90 to 100 (which is getting closer to our goal), we go up by 10. So we’ve gone up by 13 all together.
Then from 100 to 103 is three more, so we’ve gone up by 16, which means that 103 – 87 is 16.
I would be crazy to write all those steps out! – which is unfortunately what the poor child was asked to do.
But doing it quickly in your head makes a lot of sense.
As Tanton says, let’s get rid of laborious, tortured, time-wasting algorithms and examples, but let’s keep the thinking and understanding.
Click on the picture for a larger view; it’s a slightly-modified still from his video.