This is indeed a very curious book! The author takes a rather unusual and unique approach to the teaching of mathematics. His philosophical approach of turning a normal lesson into more of an enquiry where the pupils are at the heart of the learning is more in tune with today's open-questioned curriculum. Now that may not suit everyone, but it's here and is just one other tool in a teacher's box.

The book is probably aimed for upper primary or lower secondary, but could easily be adapted for other groups. It discusses numbers everywhere, from historical events to mass media trying to give pupils an understanding of the universe today. There are 28 different topics such as Pythagoras, primes, decimal point, angles, golden ratio, fractions and so on. Puzzles and problems are presented in a way to engage pupils in a way that will not turn them off or, worse, make them hate mathematics. The book might fall short for some teachers looking for full lesson plans, but that would take away the spontaneity the author is seeking to develop in a class situation.

Each chapter is divided into three main sections -” a teacher introduction for interest, a things to do section with the main ideas to try and a things to say or key words section to generate the correct atmosphere in the class. Few teachers will follow this in order, but would be better to dip in and out of the book choosing topics that are of interest to them. Otherwise their enthusiasm will be lessened which will be picked up by the class. There should be enough topics that interest most teachers to have a go and no doubt they can be developed after teaching it a couple of times. Definitely worth trying in the class.

Your answer to "What is mathematics?" has a huge bearing on how you teach mathematics. Numberverse answers the question through the subject's links with philosophy and, in the first few chapters at least, presents classroom maths as a philosophical inquiry of meaning. On that basis alone, the book is to be welcomed as providing a fresh perspective on how concepts can be introduced to primary pupils.

Numberverse has three intertwined parts. Firstly, at the start of each section, the reader is given a short introduction to the topic. This might cover tales of its historical development with which the teacher can enrich lessons; for example, we read about progressively more accurate estimates for π and the maths behind the design of arches. Alternatively, the introduction might discuss ways to teach topics in maths. Some of these are more successful than others. The suggestion for introducing fractions would overcome misconceptions I see survive into secondary classrooms; but countenancing the "adding zeros" trick for multiplying by powers of ten does not help pupils develop a conceptual understanding of place value.

The second part of each section - "things to do" - gives a precise classroom-tested script for teachers to initiate inquiry and an activity to follow. There is a diverse range of stimuli and activities: deep philosophical questions about numbers, prose and poems, standard investigations and problems to solve. My favourite is: How many squares can you form with four strips of paper and two half strips? Of course, stated in this way the problem might not provoke much curiosity in primary pupils. And that, for me, is the key message of Numberverse: "the genius in teaching is making people ready to be told" (p. 184). Draw the pupils in, arouse their curiosity through discussion, and, when they perceive a need for new knowledge, tell them.

The third part of the book will be of great interest to all inquiry teachers. Do we say the "things to say" that are suggested? Do we agree with the "key words" that form Numberverse's lexicon of inquiry teaching? Last year, I concentrated on holding the 'big picture' in focus so students could link their exploration to the purpose of the inquiry. Is that the same as the key word "anchoring"? Perhaps not, but Numberverse challenged me to consider why not.

I also found myself considering the extent to which maths can be learnt through philosophical inquiry (as opposed to mathematical inquiry). At the start of the book, Andrew Day writes that the teacher is "controlling the process completely ... but not controlling the content" (p. 18). While I would expect pupils -” certainly those in my secondary classes -” to be involved in directing the process, I also think that Numberverse, on my reading, does not hold throughout to the second part of the statement. The axiomatic nature of maths does require the teacher to control the content to an extent.

Within that "to an extent" resides the crux of classroom inquiry. The skill of the inquiry teacher lies precisely in finding the balance between eliciting students' existing knowledge and encouraging them to engage with new knowledge. It is the spirit of continuous engagement with pupils' understanding that shines through Numberverse. For that reason, the book is recommended reading for all teachers of mathematics.

Thinking and teaching mathematically can sometimes seem very black or white. Answers are usually right, or wrong and thinking creatively can seem a challenge. In fact, Day wants to make pupils feel -˜stuck' and to get children comfortable with this feeling -” a necessity to give them confidence in working their way out of problems.

The book has its roots firmly based in philosophy, with some activities very well suited for p4c sessions -” which is an interesting notion, as not many educators will have thought to incorporate numeric thinking into their philosophical sessions.

Day offers great ideas of extending this, initially simple, idea into some philosophical and mathematical thinking that can get pupils to see alternatives to their original answer. From the original answer usually given (64), you should end up with over 200 as an answer (we won't share the exact figure here. See page 133!!!). Such activities, like this one, can help develop pupils resilience -” one of the key facets advocated within the book.

We love this book. Not only does it give educators activities which you can pick up and run with, but it also is a catalyst to get children to think beyond the mathematical boundaries which they have been taught and grown accustomed to. There are activities within the book for children getting to grips with early number concepts, through to activities which have no upper-age limit. What is key, for each exercise (and teaching generally), is the questioning and considered timing to extend thinking further.

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This book certainly developed my understanding of numbers and more practical ways to bring numbers “alive” for children and young people. With the focus on promoting understanding I particularly liked the “things to do” and key word sections. These sections and other areas of the text promote ideas to get more reluctant pupils / students to enjoy maths and work more confidently with number. The focus is on helping the teacher to develop skills and strategies to challenge thinking and understanding by emphasising the importance of listening, thinking, creativity and exploratory attitudes.

Andrew Day offers the first steps to opening the minds of children to the excitement of numbers and why, through centuries of civilisation and human imagination, they have helped us to frame and shape our world.

A -˜How To' guide with a difference. This book allows me as an experienced teacher a fascinating insight into the world of maths and practical ways to help children truly understand the abstract nature of this subject. There is a glorious mixture of practical ideas, theory and anecdotes making this a very addictive read. I love the way it can be used as a dip in and out planning resource or as an absorbing cover to cover read. It guides teachers gently but firmly towards being a facilitator of learning, showing us how to use enquiry to help pupils along the path of discovery.