This paper mainly deals with the problem of short text classification. There are two main contributions. Firstly, we introduce a framework of deep uniform kernel mapping support vector machine (DUKMSVM). The significant merit of this framework is that by expressing the kernel mapping function explicitly with a deep neural network, it is in essence an explicit kernel mapping instead of the traditional kernel function, and it allows better flexibility in dealing with various applications by applying different neural network structures. Secondly, to validate the effectiveness of this framework and to improve the performance of short text classification, we explicitly express the kernel mapping using bidirectional recurrent neural network (BRNN), and propose a deep bidirectional recurrent kernel mapping support vector machine (DRKMSVM) for short text classification. Experimental results on five public short text classification datasets indicate that in terms of classification accuracy, precision, recall rate and F1-score, the DRKMSVM achieves the best performance with the average values of accuracy, precision, recall rate, and F1-score of 87.23%, 86.99%, 86.13% and 86.51% respectively compared to traditional SVM, convolutional neural network (CNN), Naive Bayes (NB), and Deep Neural Mapping Support Vector Machine (DNMSVM) which applies multi-layer perceptron for kernel mapping.