In neural network theory, we analyze two strategies for learning weights: backpropagation and random selection. The former is common in ANNs with one or more hidden layers, while the latter is becoming popular in ANNs with exactly one hidden layer and weights chosen based on the “extreme learning machine” (ELM) paradigm proposed in [4]. We show that despite the empirical success of ELM, its theoretical platform, proposed in [4], has no sound mathematical basis. We demonstrate a dataset on which ELM training and backpropagation strategies cannot obtain satisfactory accuracy. We reopen the question of a new mathematical basis with answers to the problems posed.