After a two-month break, we resumed Katie’s joint lessons with Varya. I decided to start with something simple: addition. I asked the girls to compute the sum 1+2+3+4+5+6+7+8+9. This turned out to be a harder task than I envisioned. Varya was counting diligently for a while but in the end came up with 15. “After adding a few numbers I get lost,” she confessed. Katie tried three times, each one getting a bit closer, but at the end still only got to 40.
One way that they made the task harder for themselves was by not writing down any intermediate sums, even after I encouraged them to do so. I also at some point told them that they should try rearranging the numbers to see if they can make the additions easier. They both said that they didn’t want to do that, perhaps because they didn’t understand what I meant.
Finally, I started asking them which number should be added to 1 to get 10? What about 2? 3? They got the pattern, but were disappointed that the 5 didn’t get paired up with anything. We then counted up the 10’s and added the lonely 5. They agreed with me that this calculation required very little effort, but I don’t think that they truly appreciated the idea. For that we might have to wait until they discover the formula for triangular numbers in general, but first we’ll need to polish up their arithmetic skills a bit.