Approximations of practice are opportunities for preservice teachers (PSTs) to engage in components of teaching. However, how the varying levels of complexity or authenticity of teaching create different learning opportunities for PSTs is underexplored. Forty-one studies related to approximations of mathematics teaching were reviewed to explore how approximations are embedded in PST education, how authenticity represents itself in approximations, and how mathematics teacher educators report approximations facilitating PSTs' learning. The review identified eight types of approximations used, indicating that PSTs developed aspects of specific practices such as professional noticing, conducting a meaningful mathematics discussion, as well as generic lesson planning and implementation. We propose teacher education programs consider a continuum of authenticity to capture the complexity of approximations of practice as well as the recomposition of decomposed practices. We discuss how varying levels of authenticity in approximations of practice creates different opportunities for PSTs.