The Essence of Mathematics Through Elementary Problems - cover image

Book Series


Alexandre Borovik; Tony Gardiner

Published On





  • English

Print Length

393 pages (xvi+377)


Paperback156 x 21 x 234 mm(6.14" x 0.82" x 9.21")
Hardback156 x 22 x 234 mm(6.14" x 0.88" x 9.21")


Paperback1228g (43.32oz)
Hardback1616g (57.00oz)

OCLC Number





  • PB
  • YQM
  • PBK
  • PBM
  • PBT


  • MAT000000
  • MAT004000
  • MAT002000
  • MAT012000


  • QA39.3


  • Mathematics
  • Elementary Problems
  • make sense of the world
  • mathematics beyond the classroom
  • Mental Skills
  • Arithmetic
  • Word Problems
  • Algebra
  • Geometry
  • Infinity

The Essence of Mathematics Through Elementary Problems

It is increasingly clear that the shapes of reality – whether of the natural world, or of the built environment – are in some profound sense mathematical. Therefore it would benefit students and educated adults to understand what makes mathematics itself ‘tick’, and to appreciate why its shapes, patterns and formulae provide us with precisely the language we need to make sense of the world around us. The second part of this challenge may require some specialist experience, but the authors of this book concentrate on the first part, and explore the extent to which elementary mathematics allows us all to understand something of the nature of mathematics from the inside.

The Essence of Mathematics consists of a sequence of 270 problems – with commentary and full solutions. The reader is assumed to have a reasonable grasp of school mathematics. More importantly, s/he should want to understand something of mathematics beyond the classroom, and be willing to engage with (and to reflect upon) challenging problems that highlight the essence of the discipline.

The book consists of six chapters of increasing sophistication (Mental Skills; Arithmetic; Word Problems; Algebra; Geometry; Infinity), with interleaved commentary. The content will appeal to students considering further study of mathematics at university, teachers of mathematics at age 14-18, and anyone who wants to see what this kind of elementary content has to tell us about how mathematics really works.


If you believe that offering your pupils an insight into what elementary mathematics, as encountered at school, is really about, then this book should be compulsory reading.

Gerry Leversha

"The essence of mathematics through elementary problems by Alexandre Borovik and Tony Gardiner". The Mathematical Gazette (0025-5572), vol. 105, no. 564, 2021. doi:10.1017/mag.2021.140

Full Review


I. Mental Skills

(pp. 1–50)
  • Alexandre Borovik
  • Tony Gardiner

II. Arithmetic

(pp. 51–90)
  • Alexandre Borovik
  • Tony Gardiner

III. Word Problems

(pp. 91–110)
  • Alexandre Borovik
  • Tony Gardiner

IV. Algebra

(pp. 111–168)
  • Alexandre Borovik
  • Tony Gardiner

V. Geometry

(pp. 169–282)
  • Alexandre Borovik
  • Tony Gardiner
  • Alexandre Borovik
  • Tony Gardiner


Alexandre Borovik

Professor of Pure Mathematics at University of Manchester

Tony Gardiner

Reader in Mathematics and Mathematics Education at University of Birmingham