Student Code: map function not functioning
As Edgar Bonet pointed out,  “chosenColor = 3 means 'let chosenColor take the value 3'”.  That is, your  potVal = analogRead(pot);  chosenColor = map(potVal, 0, 1023, 1, 3); sequence may very well be working, but by the time you get to Serial.println(chosenColor); you have changed chosenColor to 3.

First, the statement if(chosenColor = 1) {...} changes chosenColor to 1. Then the statement if(chosenColor = 2) {...} changes chosenColor to 2.  Finally, if(chosenColor = 3) {...} changes chosenColor to 3, just before you print out the current value of chosenColor.  

When you wish to compare two numeric values for equality, use == instead of =.