There is consensus on one point in the definition of high potential: the recognition of an intelligence quotient above 130. This definition seems to be extremely restrictive, as it is limited to a cognitive level, whereas other levels, in particular emotional and relational, are specific to what can be described as high potential in a broad sense. These different facets are expressed at school and are often a source of incomprehension for teachers and even academic malaise for pupils. Brain imaging (Gauvrit, 2014) has shown that high-potential individuals function differently in the brain, arguing for a neuro-cognitive definition of high potential. A number of terms are used to describe the specific characteristics of high potential, such as giftedness and talent, or zebra (Siaud-Fachin, 2008) for children with an atypical profile. At school, different policies have been put in place to accommodate children whose profiles are so specific that they may require special arrangements. After presenting the consensus on the definition of high potential, this paper will provide a brief overview of national policies in support of these pupils with special profiles. We will then describe our research in an adult psychotherapy clinic, based on a qualitative approach to the discourse of patients defined as high potential. Based on their experiences in school, particularly in terms of common and salient points, we will identify the specific features of this atypical profile and propose an evolving nomenclature for the concept of high potential. By incorporating Dabrowski and Piechowski’s (1977) concept of hyperexcitability, this nomenclature will be extended to include the concept of awareness developed by Perls et al. (1951). This paper aims to provide a better understanding of how so-called high-potential pupils function and to propose a new, more consensual term – awareness personality – to help teachers understand these pupils better and to help them accept their difference better.