At the special conference celebrating the 50th anniversary of the Australian Association of Mathematics Teachers (AAMT), the author was asked to speak about the challenges and opportunities in teaching mathematics as a stimulus for discussion of AAMT's "future directions". With an open invitation to be a little provocative, the author chose five challenges framed as questions: (1) Who is qualified to teach mathematics to children and young people?; (2) Is mathematics teaching an attractive and viable career?; (3) What counts as effective mathematics teaching?; (4) Who gets to learn mathematics?; and (5) How do we shape public perceptions of mathematics? The author ends her brief discussion of each challenge with another kind of question that she hopes will point us towards opportunities and directions for the next 50 years.