<p>“I took 3 math classes at a German university while in high school, which allowed me to dive directly into upper-level classes in the US. After 2 years of college in the US I have already taken as many <em>pure</em> math classes than I would have taken in my 3 years at a German university.”</p>
<p>Maybe I’m misunderstanding, but doesn’t this mean that if you hadn’t taken those classes whilst in high school, you wouldn’t have been able to take the range of pure maths classes you did in college?</p>
<p>Following on from that, the average person, who wouldn’t have had the chance to take college-level courses in high school, wouldn’t have been able to take the upper-level classes that you did, and therefore wouldn’t have been able to specialise so heavily. </p>
<p>Also, just a point about taking classes in an allied field - in the UK, it’s considered important to be able to draw links between related fields in research. That’s not considered to be breadth as such - more that you need to have a background in many related fields to in order to specialise in one, since so much research these days is collaborative anyway.</p>