Instead of correcting every grammar mistake, an easier way to improve the introduction paragraph in yesterday's lesson would be to simplify it. To simplify something, we need to focus on the message that we are trying to communicate; how can we deliver that message in the clearest, simplest way?
For example, here's a simplified version of the introduction:
Many people are confident that this century will be a positive one. I share this optimism, and there are several positive changes that I hope to see over the coming years.