Analysis sparsity and the accompanying analysis operator learning problem provide an important framework for signal modeling. Very recently, sparsifying transform learning has been put forward as an effective and new formulation for the analysis operator learning problem. In this study, we develop a new sparsifying transform learning algorithm by using the uniform normalized tight frame constraint. The new algorithm bypasses the computationally expensive analysis sparse coding step of the standard analysis operator learning algorithms. The resulting minimization problem is solved by alternating between two steps. The first step is the operator update, which comprises a least squares solution followed by a projection, and the second step is the sparse code update realized by a simple thresholding procedure. Simulation results indicate that the proposed algorithm provides improved analysis operator recovery performance when compared to a recent analysis operator learning algorithm from the literature, which uses the same uniform normalized tight frame constraint.