Helen "as I do" means "as I have to remember" (the oddities of...)
The verb Do is a dummy-word for the phrase "have to remember".
I bet you knew that.

kgcole
I think it's a good idea to memorize verbs along with the case they usually ask for.
Within a meaningful context, preferrably.

Я хочу поделиться с тобой своей радостью.
I want to share my joy with you.

делиться / поделиться / разделить / разделять с кем-либо needs instrumental and the preposition.
This is how you're refferring to the person you would like to share something with.

делиться / поделиться чем-либо needs instrumental too.
This is how you're refferring to the object to be shared.

делить / поделить / разделить / разделять что-либо needs accusative.
This is how you're refferring to the object to be shared.

That said, you can also compose something like "Я хочу разделить с тобой мою радость".
(which means almost the same as the sentence above with so miniscule difference that it isn't even worth mentioning)

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I think делить / поделить / разделить / разделять чем-либо needs instrumental because we can say something like:
Я разделила торт ножом. I divided the cake with a knife. The knife serves as a tool. That's why instrumental.