Counting Invisible Blocks

hidden_blockMarch 2, 2014
One of the first skills introduced to Robert when he was 3 or 4 was block imitation. Robert should copy the presented 3D model. I don’t remember the maximum number of blocks Robert was supposed to use. I don’t remember how well or bad he was doing. I don’t think he had ever progressed from imitating 3D models to making a structures based on 2D pictures. That was done at his private school. At home, Robert played often with Lego and built structures based on sequences of pictures. I always assisted him more or less. And of course, he usually built everything only once, so there was never any opportunity to reach mastery and being completely independent.

For the last few days, Robert was counting volumes of different 3D shapes based on their pictures in the workbook from Singapore Math, grade 3. He counted all the blocks except those he couldn’t see, as they were behind or under other blocks. I asked him to build those structures. He began but soon became confused. He saw the block in the second layer, but didn’t see the block that should be underneath it. Robert’s hand held the block in the air. That was the right position according to the picture, except it was impossible to leave the block there without the support of another block that was NOT visible in the picture.
Robert’s hesitation was priceless. The discord between the picture and the reality forced him to doubt himself.

In the last few days, he mostly build rectangular prisms. He could easily assemble them, layer by layer, based on the drawing. Although he made them without difficulties, he still had trouble finding their volumes based only on the pictures.
His first response, when presented with the drawing of the solid, was to count what he saw in the picture and not using his mind to count the invisible cubes.
Many times, after he erected the prism, I separated all the layers taking them down one by one, and then I restock them hoping Robert would remember those invisible blocks covered by the subsequent layers. Robert seemed to find the proper value of volume.
Unfortunately,with the next rectangular prism, he proceeded in the same manner, making all the same errors. He looked at the picture not to make a mental model in his mind, but to count the blocks on the visible surfaces. He was able, however, to build the rectangular prisms based on their pictures, and then count their volumes.
Today, he had difficulties with constructing more complicated models. That is why he held the block in the air and felt that something was amiss. This moment of hesitation was much more important factor in understanding the importance of invisible blocks, than the automatic recreation of the rectangular prisms.
The few seconds during which Robert’s hand hung over the space where something should be but wasn’t, provided priceless opportunity for an analytical insight. The confusion Robert felt could be the first step to recognizing the problem, and thus thinking…
I regret that I didn’t realize how important the block imitation could be, much earlier. In ABA format block imitation became rather mechanical and boring task. It was repeated over and over until given model has been replicated with 80% accuracy. There are of course good reason for that – working on attending skills could be one of them. What I haven’t realized then was that copying even simple picture might have helped Robert to replace his reliance on what is visible in the drawing to what makes sense in his mind. That the mind can fill the gap left by imperfect images.

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