A year ago, in the post https://krymarh.wordpress.com/2012/04/18/importance-of-little-words/ , I noticed that the word that impeded Robert’s learning of division of fractions was not “multiplication” , not “reciprocal” but “instead”. I used seemingly simple direction, “Instead of dividing, multiply by reciprocal.” I also concluded that this rule was much more important not as an advice on how to divide, but as an example of the meaning of the word “instead”.

A few days ago, I worked with Robert on naming angles using three letters. We, encountered the same problem we had done a few months ago, on our first try. Robert didn’t understand my direction, “Vertex has to be in the middle.” Robert knew “vertex. He could point to it without a problem. He didn’t understand “in the middle” in the context of the three letters (two of them naming points on angle’s arms and one naming the vertex).

It has to be said that part of Robert’s problem was the way I introduced this task to him. Without thinking, I just followed the problem from Saxon Math 4. The angle with a vertex A which Robert was supposed to name using three letters was one of the four angles in the quadrilateral. All the letters A, B, C, and D named vertices. The way I tried to help Robert, confused him even more.

I put the textbook aside, and drew many angles not attached to any other shapes. Now, there were three points but only one vertex. Still, it took two days of practice, before the direction, “vertex in the middle”, resulted in the correct answers. This time the culprit was, “In the middle.”.

The most important gain of those lessons was not that Robert learned how to name angles with three letters, but that he understood the concept of being in the middle in one more context.

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