I am taking an Introduction to Statistics class and need some clarity on how to find the median for a data set with an even set of numbers. It’s my understanding that the median is the midpoint in the data and that there are an equal number of data points above and below the median value. Whenever there is an even number of data points to find the median, I would just take the average of the two middle points. However, when I apply that information to the following data set (15, 11, 12, 3, 14, 17), I get an answer of 7.5 as the median (using 12 and 3 as the two middle points to average) rather than the correct answer which my answer sheet says is 13. Can you explain what I am doing wrong?

Your description of the median is correct except for one minor, but very important, point. The list must be ordered in order to compute the median.
You were given 15,11,12,3,14,17 as your data set. Ordering from least to greatest we get 3,11,12,14,15,17. Now since there is an even number of data the median is the arithmetic mean of the 3rd and 4th numbers: (12+14)/2=13.
The median would be 13.
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Most technology will order a set of numbers for you (for example Excel or a graphing utility.) Ordering your data set is a good first step: it allows you to find the range (difference between the least and greatest), compute the median,and group the numbers (for instance in a grouped frequency distribution or displayed in a stemplot, etc...)
http://mathworld.wolfram.com/StatisticalMedian.html