Mathematics teaching, and the ways in which ‘ineffective’ teaching impacts on students’ learning, has long been the topic of discussion around many a national policy table. Given that mathematics plays a central role in shaping how individuals deal with various spheres of private, social, and civil life, mathematics student outcomes have traditionally been a key rallying point for administrators and policy makers. Concerns that mathematics teaching contributes to poor students’ achievements, relative to international benchmarks, have recently put huge political pressure on education systems in a number of Western nations. Stakeholders tend to blame sitting governments at both state and national levels for not doing enough to demonstrate good or internationally comparable student performance. The current debate resembles the debates that preceded it in at least one important respect: that young people cannot do mathematics sufficiently well because, in part, teachers are not doing their job sufficiently well. The terms of the debate have prompted governments to respond with inquiries, with new initiatives, and with new policy, and the comprehensive strategy designed to boost mathematics standards in the UK is a case in point. In my own country, New Zealand, the current response consists of a suite of proposals including more public–private partnerships, increased class size, and teachers’ salaries commensurate with competence. If the proposals are implemented, inevitably, there will be some who stand to win and some who do not. Effective teaching is not necessarily a consequence of new policy. Lying at the heart of effective teaching are the knowledge and skill that an individual teacher brings to the cognitive demands of teaching. What teachers do in classrooms is very much dependent on what they know and believe about mathematics and on what they understand about the teaching and learning of mathematics (Anthony and Walshaw 2007). Successful teachers are those with both the intention and the effect to assist students to make sense of mathematics (Jaworski 2004). A teacher with merely the intention of developing student understanding will not necessarily produce the desired effect. What is clear, however, is