Can You Teach a One-Year-Old?

I think it's possible to teach a one-year-old. I'm sure that there are one-year-olds who are potty-trained, or who can speak in sentences, or who know their colors, or who at least know not to eat their colors and then use the remaining stumps to destroy the walls and furniture. MY one-year-old, however, seems to be unteachable. By me, anyway. At present. I'm struggling with everything - discipline, nap time, keeping her occupied during school time, potty training, table manners (not the complicated ones, just stuff like not throwing food everywhere) - I feel like Helen Keller's mom before she figured out sign language. (Remember that? She would just roam around the dinner table like a pet eating scraps from people's plates).
I know Grace is a good baby, and I am pretty sure, most of the time, that I'm at least an adequate caregiver, but I feel total despair about this. I've tried looking at homeschool blogs and websites about preschoolers, but they aren't helpful. Every "independent" learning activity I've come up with either fails to capture her interest or ends up in her mouth. She doesn't like sitting on my lap unless she can bang on the computer or draw on my stuff (which she can somehow distinguish from her stuff). She doesn't like to play with her own toys unless she's trapped in her room during naptime. She's discovered that ripping up her books bothers me, so she's started ripping the pages in all of her books. The only thing that keeps her quiet is the TV, and I hate using the TV.
Am I alone? Please tell me that this is a phase. Maybe it's the new baby? It started before the new baby, but maybe it's gotten worse? Has anyone figured out how to do this? Does anyone want a job as an unpaid nanny?