This evening Robert was finding distances along the routes made from two or three connected trails. He did not have any problems with finding distances by adding lengths of two or three segments dividing the entire route. When, however, he had to tell how many meters one has to make by going from A to B and back, he was lost. I made a drawing. I explained, “This is 700 meters from A to B and this is 700 meters from B to A. Robert said something, but I did not understand. So I continued. “This way is 700m and the way back is 700m.How much together?” I was sure that after hearing the magic word “together”, Robert would immediately come with addition.

Instead I saw complete confusion in Robert’s eyes. Then I heard separate words: “negative”, “zero”.

At first I did not get it. Since it was not what I expected (adding or multiplying by 2) I dismissed Robert’s answer without giving it a second thought. Besides, I was preoccupied with finding a better way to explain the problem. As I contemplated using a string to demonstrate how to measure the length of the round trip, it suddenly dawned on me.

Robert was using his knowledge of positive and negative numbers as I presented them to him on a number line. He treated both routes as opposite vectors, which had a sum of zero. Coming back to the starting point meant there was a zero shift. Just like adding 5 + (-5) =0

Robert applied his knowledge of integers in a new context.

I understood the genesis of Robert’s error. Although Robert did not understand the basic concept of length he was, nonetheless, grasping with and applying abstract concepts of positive and negative numbers as they relate to direction on the number line.

There is also the possibility that my emphasis on teaching Robert to add integers led to over generalization of the new rules.

Was that an example of teaching that backfired? Did Robert lost trust in his own analysis of the problem?

As long as I remember, Robert has never solved the problem of length of the round trip independently. So, maybe that was not a fault of my teaching. Maybe this time, he felt he had a tool to solve the problem and used it, although incorrectly.

I am not sure what to do next. Should I continue with integers on the number line and introduce the concept of absolute value as a distance from zero, or stick to the string?

I do not know what to do about that. Should I introduce absolute value or just stick to the string?

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## Jean

/ February 12, 2013Maria,

I think using the string makes sense. You can use a bike riding analogy too..ride to starbucks, then ride home would be…

Jean