I experienced a lot of chaos while teaching Robert to round up or down natural numbers or those decimals that relate to dollars and cents.

I started to teach rounding to the nearest ten approximately eight years ago. I simply used a segment of a number line, for instance, from 20 to 30. I asked Robert to find 32 and then decide if it was closer to 30 or 40. I soon stopped because I I begun to teach Robert to divide with a reminder. In a process, I used a number line in a slightly different way. I did not want Robert to be confused as to what was expected of him.

To make the matter even more baffling , in a school year 2005/2006, the teacher in a collaborative program was teaching rounding prices to the next dollar. So for $25.99 and for $ 25.01 the proper answer was $26. The authors of that part of the curriculum rightly believed that when the price is $25.30, you have to handle $26 and not $25 to the cashier. It made a lot of sense from a practical point of view, but not from mathematical understanding of the values of numbers.

In the school year 2006/2007 Robert’s teacher’s aide introduced rounding up by a set of instructions, identical to those I received in my childhood. I described it in the comment below.

I tried to continue in this manner, but Robert had a lot of problems, I was not sure how to deal with. So I stopped and waited until the teacher at school would move on to another topic. Then…. I forgot as I moved to other things.

From time to time, the need to round the numbers to estimate the result of math operation surfaced. We returned to the old approach. It seemed more visual. To round 47 to the nearest ten, we drew a line segment which had 40 and 50 written at its ends and 45 placed in the middle. Robert had to place 47 on the proper side of 45.

He had the same difficulties as in the past. I realized that he was looking for EXACT point he should place the number. That was confusing. I remedied that by drawing one circle above each half of the line segment. Now there was a space to place the number, Everything became simpler and more straightforward.

As we worked today on such problems, I realized what I am aiming for while teaching Robert.

In the past I just wanted Robert to be able to turn verbal direction into proper math operations. After reading “find average” , to follow by adding and dividing. It was a worthy goal.

That was also my first goal with rounding to the nearest ten or hundred. I wanted Robert to almost automatically follow the steps as I described them above with number 47.

Now, I consider it only a partial achievement, not the goal by itself.

**The real goal for Robert is to draw a line segment with two numbers at the ends and one in the middle… in his head. Make a mental picture and use it to solve a problem.**

I only vaguely understand why I consider this last step so important for Robert that I made it a goal worthy reaching. Is it because I want Robert to pick up the problem from a piece of paper and take it to his brain or is it because I want a proof that he can do it?

## krymarh

/ December 12, 2012The algorithm as it was taught in my childhood

1.Circle the digit you have to round to.

2.Look at the number to the right. Underline it.

3.If the underlined number is 0, 1,2,3, or 4, round down -leaving the circled number untouched. Replace all the numbers to the right of it by zeros

4.If the underlined number is 5,6,7,8,or 9, round up, increasing the circled number by one. Replace all the numbers to the right of it by zeros.