In order to understand the way children grieve, this paper will examine, the way that children grieve and how they respond at various ages, the instability of a child's behavior when they face grief and death and provide a look at methods to help children grief when they are ready.

You should never say anything like, "this paper will examine" or "in this paper, I will..."
Your thesis needs to be more concise and not so redundant. You say "the way children grieve" twice. You might try,
"In order to understand the greiving process of children, one must take into consideration how they respond to tragedy at various ages, the instability of their behavior after facing this tragedy, and the methods that can be used to help them move on with their lives."
Good luck.

Try this...

Attempting to understand the way children grieve, this paper will examine children's unstable behaviors when faced with grief and death and will also discuss methods to help make the grieving process easier.

It's shorter and more to the point, I think.

Hers is good, or try this to make the original points as solid.

In order to understand the way children grieve, this paper will examine the way that children at various ages grieve, the instability of a child's behavior when they are faced with grief or death, and provide a look at methods to help children grieve when they are ready.

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