Mathematics is a fundamental part of human knowledge and one of the central planks of the modern technological revolution. But in our enthusiasm to promote its benefits too rarely do we stop to question our intended aims of teaching mathematics and the outcomes of learning mathematics in school. In this paper the standard aims of school mathematics are acknowledged, but so too are the unexpected and unintended outcomes for some or all students. These are primarily the unplanned and unintended values, attitudes and beliefs that students develop during their years of schooling. Many of these are negative. To counter them and more generally to aim higher in school mathematics teaching I propose four additional visionary aims for mathematics that are empowering and broadening. These comprise developing higher level orientations and capabilities including mathematical confidence; mathematical creativity through problem posing and solving; social empowerment through mathematics contributing to critical citizenship; and the broader appreciation of mathematics, its nature and its key ideas.