Like Britain, the United States is alive with educational reforms. Unlike in Britain, though, school reform programmes are usually instituted at local school district level. A deeply ingrained distrust of “big government” ensures that even state-wide programmes are patchy, and difficult to mandate and deliver. That’s one reason, say the authors of Learning Policy: when state education works (by David K Cohen and Heather C Hill, Yale University Press pound;22.50), why so many reform attempts have been so misguided or mean-spirited.

Their book is a study of a project to which neither of these epithets applies: California’s state-wide attempt in the 1990s to reform the teaching and learning of mathematics in its elementary (primary) schools.

They wanted to find out not just how well it worked (to which they give a curate’s-eggish answer), but also where it worked and where it didn’t. Their key finding (not irrelevant to the English numeracy strategy) is that it was when curriculum and assessment reform went hand in hand with a rethinking of classroom practice that attainment most significantly improved.

Where teachers had extended opportunities to reflect on their own practice, they concluded, the reforms had “worked”. They give many examples of what, in classroom terms, this meant. There are good ideas here for maths teachers as well as policy-makers.

But fewer good ideas than Magdalene Lampert offers in Teaching Problems and the Problems of Teaching (Yale University Press pound;26), a gifted teacher’s day-by-day analysis of one year’s mathematics lessons with a very mixed class of 10-year-olds in upstate Michigan.

This is a major research project (every lesson observed and videotaped, a daily teaching journal, structured field notes, a huge assessment database) and a marvellous insight into the teacher’s craft.

Lampert’s central themes are the multi-dimensional complexity of teaching and learning, and the richness of the strategies that can be deployed to resolve them. Her chapter headings (they include “Teaching to Establish a Classroom Culture”, “Teaching while Preparing a Lesson”, “Teaching while Students Work Independently”, “Teaching the Whole Class”, “Teaching Closure”) convey something of her style and her approach. One chapter is a transcript of a whole-class discussion lesson: as you read it, you can see the learning happening.

This is a well-crafted, modest, richly pleasurable, even gripping, book, that says more about the challenges and pleasures of teaching (and learning) than a host of manuals could. You don’t need to be a mathematics teacher to enjoy it, although once you have read it, you may wish you were. Excellent.

One of Lampert’s key messages is that appropriate assessment is at the heart of all good teaching. That is very much the theme also of Knowing What Students Know: the science and design of educational assessment (National Academy Press pound;26.95). This is a major report commissioned by the prestigious National Research Council. James Pellegrino, who chaired the research committee, is among the editors of this publication, which is aimed at policy-makers rather than practising teachers. Interesting for all, however, as it strikes powerful and necessary blows against the politicians and the media who still, in the UK as well as the US, yearn for “simple pencil-and-paper tests” as measures of achievement.

Simple assessment, it argues, is a contradiction in terms. The most important function of assessment is to make students’ thinking clear to their teachers and to themselves; there is a fascinating section here on identifying the “bugs” that block a student’s learning. But there are many other functions of assessment and ways to use the information. The key principle is that assessment, whether at classroom or national level, should be based on how people learn and should be valid for its purpose. Little current practice in the US meets these two criteria, the report suggests; it provides extensive illustrations of alternatives that would.

It sounds dry, but in the context of the current uncertainty in England and Wales about GCSE and A-levels, it makes for stimulating and timely reading. Why, for example, is the A-level “gold standard” so important?

In the terms of this volume, our own assessment debate is still at key stage 1. Knowing What Students Know could be a powerful contribution.