Noriko,
Thanks for sharing your story. You're students are very lucky to get so much individual attention from you. I'm sure you'll discover which games make them -and you- happiest over time.
How did you explain to your little one about kindess vs. winning? Is her English level very high, or do you speak her first language? I wish I was able to explain everything to my kids directly about the behaviour I prefer in class. Since my Korean skills aren't
that good, I either ask a Korean person to help me, or simply try to model good behaviour. Needless to say, there are plenty of times in my classes when everybody feels ambiguous about what is going on.
Drawing dictation and act-it-out activities with storybooks are two other kinds of things I like to do with kindergarten classes that don't involve competition.
Let me know what activities you find work best with your group.
All the best,
Karen