Some 50 years ago, the business management guru Kurt Lewin wrote: “There’s nothing so practical as good theory.” A catchy adage, but is it just provocative? What real use are theories about teaching and learning to real classroom teachers?

Learning Mathematics is as good a place as any to begin to find out. It is not a book of fluffy bright ideas, but a treasure chest of big ideas - not just for tomorrow’s lesson, but for a lifetime of them.

A taste of one recurrent theme - tensions - captured in the third and final section of the book, is of more use here than a light sprinkling of chapter titles. The post-war years have seen the emergence of “understanding” as a major goal in every mathematics classroom. No longer content for students to know that, or to know how, we want them to know why.

Thus “constructivism” comes to rear its head as a way of thinking and talking about what it means for children to understand, or to know anything at all.

In a nutshell, constructivism asserts that - essentially and literally - knowledge is what we make of it, and each individual builds his or her knowledge in a unique way. Every time teachers ask a child to explain a mental strategy, or to name a diamond shape (or is it a square?), they demonstrate constructivist belief in action.

Barbara Jaworski heightens our awareness of one inescapable tension inherent in constructivism as worked out by teachers in the classroom.

This is the tension between recognising and valuing the many kinds of understandng constructed by students, while recognising the “official” forms of knowledge demanded by the curriculum and examination system.

Must it be inevitable that teachers are driven back to “telling” as the only means of teaching that will meet such demands? Terry Wood and Tammy Turner-Vorbeck insist that for students to understand, teachers must resist the inclination to tell, and learn to teach in ways that go against their natural tendencies.

John Mason discusses the “didactic tension” - that the more explicit the teacher is about the behaviour demanded of students (what they say, what they write), the easier it is for the student to display that behaviour without attaching meaning to it. This tension drives us to hold back from challenging students in the tasks we set for them, because routine tasks seem to be more comfortable for teacher and student. Yet, as Piaget teaches us with his notion of accommodation, significant learning moments are rarely comfortable.

Each chapter has its fair share of “isms” and “ologies” to be grappled with. Jargon, if you will, yet John Mason is at pains to point out that we need such labels to entertain and work with ideas of all kinds in professional communities.

Given the lack of any clear theory or philosophy of learning in the national numeracy strategy, teachers will need to find - indeed, to construct - their own. There are no neat, universal solutions, but this book offers the prospect of engaging with and living in creative tension in the classroom.

Tim Rowland lectures in mathematics education at Homerton College, University of Cambridge