Sam and I put together the Decanomial Square over the past 2 weeks. (Yes, it took 2 weeks with all the sick days in between.) As we went along, I asked her to use her Multiplication Finger Chart and to count the beads for some multiplication problems. She checked her answers on her new calculator. (She trusts that calculator more than she trusts her multiplication control chart, and definitely more than she trusts me.) I never saw any kind of “aha” light bulb, though. Then we finished the Decanomial Square and I didn’t want to put it away, so I thought I’d introduce her to the concept of “squares,” even though I didn’t think she’d really get it.
Here is the built Decanomial Square:
Since she didn’t do the greatest job making things neat and square, I was a little lost about how I’d talk about squares. And she was getting kind of tired of doing problems and counting the beads. I ended up pulling out a sheet of graph paper and drawing squares and rectangles to remind her of the difference between them. Again, I couldn’t really tell if she was getting anything out of it. I let her color in the shapes. I counted the length of the sides. Nothing seemed to be getting through. Then, I tried counting the graph paper boxes within the shapes (which we called “units”), just like we’d count the beads from the Decanomial Square to get the area of each shape (although I didn’t introduce the word “area”). I wrote the numbers in the boxes, left to right, top to bottom (consistent with how we were working the Decanomial Square). Then we counted the appropriate beads and got the same answer. Then she did the multiplication problem on her calculator and got the same answer. Finally, I got my “aha” moment. Somehow, that made it all clear to her and Sammy was thrilled! Again, it took doing two exercises at the same time, one just a level more abstract than the other, to get the concept through.