“Can you refresh my memory?”
You're talking with your friend from high school about other people who used to go to school with you. Your friend asks if you remember a certain person, but you don't really remember that person. You want your friend to give some more information about that person to help you remember, so you ask this.
Can you refresh my memory?
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This phrase means to remind someone of something that they used to know. You "refresh" someone's memory by giving them more information or reviewing things that they should already know.
You use "refresh ___'s memory" like this:
I couldn't remember what it was called. James had to refresh my memory.
Can you refresh my memory as to how to fill out this expense request form?
This phrase can be used in both casual and polite conversation.