January 18, 2016

For a few months now, Robert and I have been occupied with adding integers. He was busy counting steps forward or backward depending on the sign of the number while I was searching for ways to help Robert generalize the skill to those numbers that didn’t fit on the line with its twenty five units starting with negative 12 and ending with 12.

Although I presented Robert with formulas for three cases of adding integers: (both negative, one negative and one positive when the negative one has larger absolute value, and one negative and one positive when the positive one has larger absolute value), I had doubts that Robert can remember and properly apply the rules expressed through overly long sentences.

So I tried something else. Instead of placing numbers from -12 to 12 on the number line I placed tens starting with -120 and ending with 120. As long as I asked Robert to add -30+20 or -40+(-50) Robert, counting by tens. moved appropriately one way or another.

I hoped that if Robert knows that -30+20=-10 then he might notice that -30+ 17= -13 or that -27+20=-7.

But Robert didn’t notice.

I had to divide each segment into 10 smaller ones – millimeter long at best, so Robert could keep counting.

So he still relied on mechanically counting and not on doing the proper operations in his head.

Now, I just draw a line with a zero in its center. Above this line I drew a few arrows to the right with plus signs above them and a few arrows to the left with a minus drawn above them.

Below I write a few problems starting with the same number.

-7+8= -7+(-8)= -7+4= -7+7=

Robert has to point to the place where -7 could be (just any place to the left of zero) and then move backward or forward and come with an answer.

He cannot count by one as there is nothing to be counted on the number line, he has to do all those operations in his head deciding by himself if the answer is moving past zero in one or another direction or if it moves farther away from zero. Thus, he has to add or subtract absolute values and decide on which side of zero the answer is placed.

We do just a few operations like that every day. Robert still has problems with going in the right directions. When, however, he chooses the correct direction he quickly comes with the proper answer.

The fact that Robert chooses too often the arrow with the wrong direction is a result of the habit, he has never conquered in general (he manages it in some specific situations but not in all). Robert answers too quickly before applying all the information he gets. To be blunt – Robert answers without thinking.

If I slow him down, stopping him from automatic answer, his performance improves. That means just covering the problem for a second, asking him to repeat the operation from his memory.

Why is it so important for me to have Robert dto those operations correctly? Of course, I want Robert to know more and understand more. But what motivates me mostly is my wish that Robert learns to generalize skills – from the numbers he can see on number line to numbers he can see only in his mind.