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from PT35/sec4/Q1

[People who are very good at manipulating symbols do not necessarily have any mathematical understanding]

What does this mean?

1. Does it simply mean that people who are very good at manipulating symbols do not have any mathematical understanding?

or

2. People who are very good at manipulating symbols do not always have any mathematical understanding.

English is my second language so help me out here.

I don't get what "do not necessarily~have any" means in this sentence.

Option 2

It is closer to 2.
It is saying that being very good at manipulating symbols is not sufficient to draw a conclusion regarding ones mathematical understanding. They could have no understanding, some understanding, full understanding but we don't have enough information to know which.

May not always would be better than do not always. It's possible they may always have mathematical understanding, but we don't know for sure.

thanks a lot you two!

That cleared it up!