ANSWER:

I got the answer, and it was simple! Here is my revised code:

def sumDivisors(num, divisorSums={}):

    if num in divisorSums:
        return divisorSums[num]

    total = 0

    if num == 1:
        return 0

    for i in xrange(num/2, 0, -1):
            if not num % i:
                total += i

    divisorSums[num] = total

    return total

I needed to remove the first conditional statement. Because that would make every value return 1.

Tetramputechture from http://stackoverflow.com/questions/16640738