Does one need a master's in math before taking a PhD in pure math?
It depends on the country. 

In the US, it is frequent that students go from undergrad right into their PhD. While there, some do get a Master's, but as a side effect of coursework for the Ph.D. Though, I myself did get a Master's first. 

I believe it is more common in Europe to get a Master's first due to the shortened Ph.D. process there.