The Essence of Mathematics Through Elementary Problems

The Essence of Mathematics Through Elementary Problems Alexandre Borovik and Tony Gardiner
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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, Mathematical Gazette, October 2021

It is clear in the text that the essence of mathematics is in the doing of mathematics rather than in particular subjects or concepts, which is not to say that concepts and ideas are not important. [... This] is certainly not a revolutionary claim [...].  And yet, I was often surprised by how this book made me rethink pretty elementary, even standard problems, and how it often led me to deeper matters in a very natural way. [... S]uch elementary problems have this power if they are set in the right context and, I might add, if they land in the lap of the right person. [...] [D]eeper lessons are [...] embedded in these very elementary problems. [... O]ne person might get more out of the book than another, but anyone reading it reflectively will be bound to get something out of it.  [...]  Who are the intended readers of this book? [... Anyone] interested in mathematics [... especially] mathematics educators.
-  Michael N. Fried (2020): From another tribe of mathematics educators, Mathematical Thinking and Learning,available here.

This book is a marvel!  From the outset [it] sets out its stall ... as a tool for developing ... rich mathematical understanding and experiences for students ... throughout their upper school mathematics. ... The book is a key tool in developing student conceptions of mathematics ... .  [I]n addition the problems are ideal for teachers wishing to deepen [their own] understanding through a mixture of the novel and the classic.  The problems have already found their way into both my classroom and my departmental meetings, but ... this is not a book it's possible to read without digging into the maths yourself!
- Mathematics in School, May 2020 


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.



The Essence of Mathematics Through Elementary Problems
Alexandre Borovik and Tony Gardiner | June 2019
OBP Series in Mathematics, vol. 3 | ISSN: 2397-1126 (Print); 2397-1134 (Online)
389 pp. | 6.14" x 9.21" (234 x 156 mm)
ISBN Paperback: 9781783746996
ISBN Hardback: 9781783747009
ISBN Digital (PDF): 9781783747016
DOI: 10.11647/OBP.0168
BIC subject codes: PB (Mathematics) | YQM (Educational: Mathematics and numeracy) | PBK (Calculus and mathematical analysis) | PBM (Geometry) | PBT (Probability and statistics); BISAC subject codes: MAT034000 (MATHEMATICS / Mathematical Analysis), MAT005000 (MATHEMATICS / Calculus), MAT012000, (MATHEMATICS / Geometry / General), MAT029000 (MATHEMATICS / Probability & Statistics / General); OCLC number: 1107447359

Preface
About this text

  1. Mental Skills
    1. Mental arithmetic and algebra
      1. Times tables
      2. Squares, cubes, and powers of 2
      3. Primes
      4. Common factors and common multiples
      5. The Euclidean algorithm
      6. Fractions and ratio
      7. Surds
    2. Direct and inverse procedures
      1. Factorisation
    3. Structural arithmetic
    4. Pythagoras' Theorem
      1. Pythagoras' Theorem, trig for special angles, and CAST
      2. Converses and Pythagoras' Theorem
      3. Pythagorean triples
      4. Sums of two squares
    5. Visualisation
    6. Trigonometry and radians
      1. Sine Rule
      2. Radians and spherical triangles
      3. Polar form and sin(A+B)
    7. Regular polygons and regular polyhedra
      1. Regular polygons are cyclic
      2. Regular polyhedra
    8. Chapter 1: Comments and solutions
  2. Arithmetic
    1. Place value and decimals: basic structure
    2. Order and factors
    3. Standard written algorithms
    4. Divisibility tests
    5. Sequences
      1. Triangular numbers
      2. Fibonacci numbers
    6. Commutative, associative and distributive laws
    7. Infnite decimal expansions
    8. The binary numeral system
    9. The Prime Number Theorem
    10. Chapter 2: Comments and solutions
  3. Word Problems
    1. Twenty problems which embody "3 - 1 = 2"
    2. Some classical examples
    3. Speed and acceleration
    4. Hidden connections
    5. Chapter 3: Comments and solutions
  4. Algebra
    1. Simultaneous linear equations and symmetry
    2. Inequalities and modulus
      1. Geometrical interpretation of modulus, of inequalities, and of modulus inequalities
      2. Inequalities
    3. Factors, roots, polynomials and surds
      1. Standard factorisations
      2. Quadratic equations
    4. Complex numbers
    5. Cubic equations
    6. An extra
    7. Chapter 4: Comments and solutions
  5. Geometry
    1. Comparing geometry and arithmetic
    2. Euclidean geometry: a brief summary
    3. Areas, lengths and angles
    4. Regular and semi-regular tilings in the plan
    5. Ruler and compasses constructions for regular polygons
    6. Regular and semi-regular polyhedra
    7. The Sine Rule and the Cosine Rule
    8. Circular arcs and circular sectors
    9. Convexity
    10. Pythagoras' Theorem in three dimensions
    11. Loci and conic sections
    12. Cubes in higher dimensions
    13. Chapter 5: Comments and solutions
  6. Infinity: recursion, induction, infinite descent
    1. Proof by mathematical induction
    2. 'Mathematical induction' and 'scientific induction'
    3. Proof by mathematical induction II
    4. Infinite geometric series
    5. Some classical inequalities
    6. The harmonic series
    7. Induction in geometry, combinatorics and number theory
    8. Two problems
    9. Infinite descent
    10. Chapter 6: Comments and solutions
Alexandre Borovik is a Professor of Pure Mathematics at the University of Manchester, United Kingdom. His research interests include the philosophy of mathematics, mathematics education, and popularisation of mathematics.

Tony Gardiner held the position of Reader in Mathematics and Mathematics Education at the University of Birmingham until December 2011. In 2011, Tony was elected Education Secretary of the London Mathematical Society.