Or maybe. Why doesn't your mother claim it on her taxes?
To claim the Dependent Care Credit, you must provide the SS# of the person you paid to care for your child, on your tax return (form 2441). The IRS will be looking for that income on your mother's return. If she has enough income to be required to file a tax return, she must include that income.
Furthermore, if she does the babysitting in her home, rather than yours*, she must treat the income as self employment income and pay social security and medicare (FICA) tax as well as income tax. In that case, she is required to file a tax return with as little as $400 of self employment income.
*If she sits in your home, she is considered a household employee, rather than self employed. You parent is exempt form FICA, in that situation.