These readable books by authors with substantial experience of teaching primary mathematics on PGCE courses give a definite sense of trying to communicate with students weighed down with “audit anxiety”. Both draw freely on research and practice in the teaching of mathematics.

The scrutiny of trainee teachers’ own knowledge of the core subjects in the new curriculum for Initial Teacher Training has spawned a number of publications claiming to help students sharpen their mathematical understanding. Mathematical Knowledge for Primary Teachers is among the best of them, and I wish I had been in a position to recommend it to my current students at the start of their course.

All trainees must have a pass in GCSE mathematics, but the authors argue that a different kind of mathematical knowledge is needed for teaching and that their book is not a GCSE revision text. The emphasis is on understanding and making connections between topics and the exposition is friendly and generally very clear.

The discussion of various methods of whole-number calculation might challenge the concentration of the lone reader, but that is largely inherent in the subject-matter. What matters is that it is included, so that my students can follow up a great deal of my (inevitably incomplete) teaching.

Anne Cockburn’s book is also one that I shall want my students to know about. The “insight” of the title, which is analysed in Chapter 4, is the intended outcome of “brainstorming” a topic before and during teaching it. As well as understanding the mathematics itself, the teacher needs awareness of potential psychological obstacles to learning.

This general approach is illustrated in detail in chapters on subtraction, place value, time and shape. The author frequently draws from her own research: to remind us, for example, that a significant part of children’s school-time agenda is not about “learning”, but what John Holt called “getting through the day” as pleasantly and unobtrusively as possible.

Cockburn has a sympathetic but direct style, neither stuffy nor patronising. I was unsure about the book’s subtitle: perhaps the requirement to recognise children’s errors and misconceptions in the new training “standards” led me to expect a different emphasis. In the end, I liked the book for its refreshing style, its usefulness and the inherent interest of its subject matter.

Tim Rowland lectures in primary mathematics education at the Institute of Education, University of London.