How many ways to come up with four teammates? A Math teacher has to choose 4 people for a competition. If there are 6 boys and 6 girls, and the teacher must select 2 boys and 2 girls, how many ways are there?
I came up with ${4 \choose 2} \cdot {4 \choose 2}$. This is because there are 4 spots and there must be 2 from each gender. Is this the right method? 

"This is because there are 4 spots and there must be 2 from each gender."
Suggestion.  The first important thing in solving a combinatorics problem involving selection is to make sure you know what you are selecting.
The above quotation seems to indicate that you are selecting spots.  But you're not - you know exactly how many spots there are.  You are selecting the people to go into those spots.
Specifically, you are choosing boys to go into two of the spots, and girls for the other two spots.
The next thing you should do is to determine whether or not repetitions are allowed in your selection, and whether or not order is important.  The answers to these questions may be stated in the problem, but often you will find that you are expected to work them out from context.
Finally, it is always good practice to explain clearly what you are doing and not just to give the answer.  So I would suggest the following as a model answer to the first part of the question.

We have to choose $2$ boys from $6$ options.  We cannot choose the same boy twice, and the order is not important.  So the number of possible selections is $\binom62$.

