Recent concerns about a shortage of capacity for statistical and numerical analysis skills among social science students and researchers have prompted a range of initiatives aiming to improve teaching in this area. However, these projects have rarely re-evaluated the content of what is taught to students and have instead focussed primarily on delivery. The emphasis has generally been on increased use of complex techniques, specialist software and, most importantly in the context of this paper, a continued focus on inferential statistical tests, often at the expense of other types of analysis. We argue that this "business as usual" approach to the content of statistics teaching is problematic for several reasons. First, the assumptions underlying inferential statistical tests are rarely met, meaning that students are being taught analyses that should only be used very rarely. Secondly, all of the most common outputs of inferential statistical tests--p-values, standard errors and confidence intervals--suffer from a similar logical problem that renders them at best useless and at worst misleading. Eliminating inferential statistical tests from statistics teaching (and practice) would avoid the creation of another generation of researchers who either do not understand, or knowingly misuse, these techniques. It would also have the benefit of removing one of the key barriers to students' understanding of statistical analysis. [For a response to this article, "A Response to White and Gorard: against Inferential Statistics: How and Why Current Statistics Teaching Gets It Wrong," see EJ1152509. For the rejoinder, "Still against Inferential Statistics: Rejoinder to Nicholson and Ridgway," see EJ1152510.]